Apparatus and method for generating audio subband values and apparatus and method for generating time-domain audio samples

ABSTRACT

An embodiment of an apparatus for generating audio subband values in audio subband channels includes an analysis windower for windowing a frame of time-domain audio input samples being in a time sequence extending from an early sample to a later sample using an analysis window function including a sequence of window coefficients to obtain windowed samples. The analysis window function includes a first number of window coefficients derived from a larger window function including a sequence of a larger second number of window coefficients, wherein the window coefficients of the window function are derived by an interpolation of window coefficients of the larger window function. The apparatus further includes a calculator for calculating the audio subband values using the windowed samples.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a U.S. National Phase entry of PCT Application No.PCT/EP2007/009200 filed 23 Oct. 2007, which claims priority to U.S.Provisional Patent Application No. 60/862,954 filed 25 Oct. 2006.

BACKGROUND OF THE INVENTION

Embodiments of the present invention relate to an apparatus and methodfor generating audio subband values, an apparatus and a method forgenerating time-domain audio samples and systems comprising any of theaforementioned apparatuses, which can for instance be implemented in thefield of modern audio encoding, audio decoding or other audiotransmission-related applications.

Modern digital audio processing is typically based on coding schemeswhich enable a significant reduction in terms of bit rates, transmissionbandwidths and storage space compared to a direct transmission orstorage of the respective audio data. This is achieved by encoding theaudio data on the sender side and decoding the encoded data on thereceiver side before, for instance, providing the decoded audio data toa listener or to a further signal processing.

Such digital audio processing systems can be implemented with respect toa wide range of parameters, typically influencing the quality of thetransmitted or otherwise processed audio data, on the one hand, andcomputational efficiency, bandwidths and other performance-relatedparameters, on the other hand. Very often, higher qualities necessitatehigher bit rates, an increased computational complexity and a higherstorage requirement for the corresponding encoded audio data. Hence,depending on the application in mind, factors as allowable bit rates, anacceptable computational complexity and acceptable amounts of data haveto be balanced with a desirable and achievable quality.

A further parameter, which is especially important in real-timeapplications such as a bi-directional or a mono-directionalcommunication, the delay imposed by the different coding schemes mayalso play an important role. As a consequence, the delay imposed by theaudio encoding and decoding poses a further constraint in terms of thepreviously mentioned parameters when balancing the needs and the costsof different coding schemes having a specific field of application inmind. As such digital audio systems can be applied in may differentfields of applications ranging form ultra-low quality transmission to ahigh-end-transmission, different parameters and different constraintsare very often imposed on the respective audio systems. In someapplications, a lower delay may for instance necessitate a higher bitrate and, hence, an increased transmission bandwidth compared to anaudio system with a higher delay, as comparable quality level.

However, in many cases, compromises may have to be taken in terms ofdifferent parameters such as bit rate, computational complexity, memoryrequirements, quality and delay.

SUMMARY

According to an embodiment, an apparatus for generating audio subbandvalues in audio subband channels may have: an analysis windower forwindowing a frame of time-domain audio input samples being in a timesequence extending from an early sample to a later sample using ananalysis window function including a sequence of window coefficients toacquire windowed samples, the analysis window function including a firstnumber of window coefficients derived from a larger window functionincluding a sequence of a larger second number of window coefficients,wherein the window coefficients of the window function are derived by aninterpolation of window coefficients of the larger window function; andwherein the second number is an even number; and a calculator forcalculating the audio subband values using the windowed samples.

According to another embodiment, an apparatus for generating time-domainaudio samples may have: a calculator for calculating a sequence ofintermediate time-domain samples from audio subband values in audiosubband channels, the sequence including earlier intermediatetime-domain samples and later time-domain samples; a synthesis windowerfor windowing the sequence of intermediate time-domain samples using asynthesis window function including a sequence of window coefficients toacquire windowed intermediate time-domain samples, the synthesis windowfunction including a first number of window coefficients derived from alarger window function including a sequence of a larger second number ofwindow coefficients, wherein the window coefficients of the windowfunction are derived by an interpolation of window coefficients of thelarger window function; and wherein the second number is even; and anoverlap-adder output stage for processing the windowed intermediatetime-domain samples to acquire the time-domain samples.

According to another embodiment, a method for generating audio subbandvalues in audio subband channels may have the steps of: windowing aframe of time-domain audio input samples being in a time sequenceextending from an early sample to a later sample using an analysiswindow function to acquire windowed samples, the analysis windowfunction including a first number of window coefficients derived from alarger window function including a sequence of a larger second number ofwindow coefficients, wherein the window coefficients of the windowfunction are derived by an interpolation by window coefficients of thelarger window function; and wherein the second number is an even number;and calculating the audio subband values using the windowed samples.

According to another embodiment, a method for generating time-domainaudio samples may have the steps of: calculating a sequence ofintermediate time-domain samples from audio subband values in audiosubband channels, the sequence including earlier intermediatetime-domain samples and later intermediate time-domain samples;windowing the sequence of intermediate time-domain samples using asynthesis window function to acquire windowed time-domain samples, thesynthesis window function including a first number of windowcoefficients derived from a larger window function including a sequenceof a larger second number of window coefficients, wherein the windowcoefficients of the window function are derived by an interpolation ofwindow coefficients of the larger window function; and wherein thesecond number is even; and overlap-adding the windowed time-domainsamples to acquire the time-domain samples.

Another embodiment may have a program with a program code for executinga method for generating audio subband values in audio subband channels,the method including: windowing a frame of time-domain audio inputsamples being in a time sequence extending from an early sample to alater sample using an analysis window function to acquire windowedsamples, the analysis window function including a first number of windowcoefficients derived from a larger window function including a sequenceof a larger second number of window coefficients, wherein the windowcoefficients of the window function are derived by an interpolation bywindow coefficients of the larger window function; and wherein thesecond number is an even number; and calculating the audio subbandvalues using the windowed samples, when running on a processor.

Another embodiment may have a program with a program code for executinga method for generating time-domain audio samples, the method including:calculating a sequence of intermediate time-domain samples from audiosubband values in audio subband channels, the sequence including earlierintermediate time-domain samples and later intermediate time-domainsamples; windowing the sequence of intermediate time-domain samplesusing a synthesis window function to acquire windowed time-domainsamples, the synthesis window function including a first number ofwindow coefficients derived from a larger window function including asequence of a larger second number of window coefficients, wherein thewindow coefficients of the window function are derived by aninterpolation of window coefficients of the larger window function; andwherein the second number is even; and overlap-adding the windowedtime-domain samples to acquire the time-domain samples, when running ona processor.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will be detailed subsequentlyreferring to the appended drawings, in which:

FIG. 1 shows a block diagram of an embodiment of an apparatus forgenerating audio subband values;

FIG. 2 a shows a block diagram of an embodiment of an apparatus forgenerating time-domain audio samples;

FIG. 2 b illustrates a functional principle according to an embodimentof the present invention in the form of an apparatus for generatingtime-domain samples;

FIG. 3 illustrates the concept of interpolating window coefficientsaccording to an embodiment of the present invention;

FIG. 4 illustrates interpolating window coefficients in the case of asine window function;

FIG. 5 shows a block diagram of an embodiment of the present inventioncomprising a SBR decoder and a SBR encoder;

FIG. 6 illustrates the delay sources of a SBR system;

FIG. 7 a shows a flowchart of an embodiment of a method for generatingaudio subband values;

FIG. 7 b illustrates a step of the embodiment of the method shown inFIG. 7 a;

FIG. 7 c shows a flowchart of an embodiment of a method for generatingaudio subband values;

FIG. 8 a shows a flowchart of a comparative example of a method forgenerating time-domain samples;

FIG. 8 b shows a flowchart of a comparative example of a method forgenerating time-domain samples;

FIG. 8 c shows a flowchart of an embodiment of a method for generatingtime-domain samples;

FIG. 8 d shows flowchart of another embodiment of a method forgenerating time-domain samples;

FIG. 9 a shows a possible implementation of a comparative example of amethod for generating audio subband values;

FIG. 9 b shows a possible implementation of an embodiment of a methodfor generating audio subband values;

FIG. 10 a shows a possible implementation of a comparative example of amethod for generating time-domain samples;

FIG. 10 b shows a further possible implementation of an embodiment of amethod for generating time-domain samples;

FIG. 11 shows a comparison of a synthesis window function according toan embodiment of the present invention and a sine window function;

FIG. 12 shows a comparison of a synthesis window function according toan embodiment of the present invention and a SBR QMF prototype filterfunction;

FIG. 13 illustrates the different delays caused by the window functionand the prototype filter function shown in FIG. 12;

FIG. 14 a shows a table illustrating different contributions to thedelay of a conventional AAC-LD+SBR codec and an AAC-ELD codec comprisingan embodiment of the present invention;

FIG. 14 b shows a further table comprising details concerning the delayof different components of different codecs;

FIG. 15 a shows a comparison of a frequency response of an apparatusesbased on a window function according to an embodiment of the presentinvention and an apparatus based on a sine window function;

FIG. 15 b shows a close-up of the frequency response shown in FIG. 15 a;

FIG. 16 a shows a comparison of the frequency response of 4 differentwindow functions;

FIG. 16 b shows a close-up of the frequency responses shown in FIG. 16a;

FIG. 17 shows a comparison of a frequency response of two differentwindow functions, one window function according to the present inventionand one window function being a symmetric window function;

FIG. 18 shows schematically the general temporal masking property of thehuman ear; and

FIG. 19 illustrates a comparison of an original audio time signal, atime signal generated based on HEAAC codec and a time signal based oncodec comprising an embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

FIGS. 1 to 19 show block diagrams and further diagrams describing thefunctional properties and features of different embodiments ofapparatuses and methods for generating audio subband values, ofapparatuses and methods for generating time-domain samples and systemscomprising at least one of the aforementioned apparatuses or methods.However, before describing a first embodiment of the present inventionin more detail, it should be noted that embodiments of the presentinvention can be implemented in hardware and in software. Hence,implementations described in terms of block diagrams of hardwareimplementations of respective embodiments can also be considered asflowcharts of an appropriate embodiment of a corresponding method. Also,a flowchart describing an embodiment of the present invention can beconsidered to be a block diagram of a corresponding hardwareimplementation.

In the following, implementations of filterbanks will be described,which can be implemented as an analysis filterbank or a synthesisfilterbank. An analysis filterbank is an apparatus for generating audiosubband values in audio subband channels based on time-domain audio(input) samples being in a time sequence extending from an early sampleto a later sample. In other words, the term analysis filterbank can besynonymously used for an embodiment of the present invention in the formof an apparatus for generating audio subband values. Accordingly, asynthesis filterbank is a filterbank for generating time-domain audiosamples from audio subband values in audio subband channels. In otherwords, the term synthesis filterbank can be used synonymously for anembodiment according to the present invention in the form of anapparatus for generating time-domain audio samples.

Both, an analysis filterbank and a synthesis filterbank, which are alsoreferred to summarizing as filterbanks, may for instance be implementedas modulated filterbanks. Modulated filterbanks, examples andembodiments of which will be outlined in more detail below, are based onan oscillations having frequencies which are based on or derived fromcenter frequencies of corresponding subbands in the frequency-domain.The term “modulated” refers in this context to the fact that theaforementioned oscillations are used in context with a window functionor a prototype filter function, depending on the concrete implementationof such a modulated filterbank. Modulated filterbanks can in principlebe based on real-valued oscillations such as a harmonic oscillation(sine-oscillation or cosine-oscillation) or corresponding complex-valuedoscillations (complex exponential oscillations). Accordingly, themodulated filterbanks are referred to as real modulated filterbanks orcomplex filter modulated filterbanks, respectively.

In the following description, embodiments of the present invention inthe form of complex modulated low-delay filterbanks and real modulatedlow-delay filterbanks and corresponding methods and softwareimplementations will be described in more detail. One of the mainapplications of such modulated low-delay filterbanks is an integrationinto a low-delay spectral band replication system (SBR), which currentlyis based on using a complex QMF filterbank with a symmetric prototypefilter (QMF=Quadrature Mirror Filter).

As will become apparent in the framework of the present description, animplementation of low-delay filterbanks according to embodiments of thepresent invention will provide the advantage of an improved trade-offbetween computational complexity, frequency response, temporal noisespreading and (reconstruction) quality. Furthermore, an improvedtrade-off between delay and reconstruction quality is achievable basedon an approach to make use of so-called zero-delay techniques to extendthe filter impulse response of the respective filterbanks withoutintroducing additional delay. A lower delay at a predefined qualitylevel, a better quality at a predefined delay level or a simultaneousimprovement of both the delay and the quality, can be achieved byemploying an analysis filterbank or a synthesis filterbank according toan embodiment of the present invention.

Embodiments of the present invention are based on the finding that theseimprovements can be achieved by employing an interpolation scheme toobtain a window function having a first number of window coefficientsbased on a window function having a larger second number of windowcoefficients. By employing an interpolation scheme, an improveddistribution of energy values of the window coefficients of the windowfunctions can be achieved. This leads in many cases to an improvedaliasing level and an improvement with respect to the audio quality. Forinstance, when the larger window function comprises an even number ofwindow coefficients, an interpolation scheme may be useful.

The computational complexity increases only slightly by employing aninterpolation scheme. However, this slight increase is not onlyoutweighed by the improvement concerning the quality but also by theresulting savings concerning the reduced memory usage when comparing thesituation with two separate window functions being stored independently.While the interpolation can be carried out in one or a few cycles of theclock signal of a processor in an implementation, in many cases leadingto an insignificant delay and increased computational complexity, theadditional memory requirement may be extremely important in manyapplications. For instance, in the case of mobile applications, thememory may be limited, especially when long window functions having asignificant number of window coefficients are employed.

Moreover, embodiments according to the present invention can be used incontext with a new window function for any of the two filterbanksdescribed above, further improving the aforementioned trade-offs. Thequality and/or the delay can further be improved in the case of ananalysis filterbank by employing an analysis window function comprisinga sequence of window coefficients, which comprises a first groupcomprising a first consecutive portion of the sequence of windowcoefficients and the second group of window coefficients comprising asecond consecutive portion of the sequence of window coefficients. Thefirst portion and the second portion comprise all window coefficients ofthe window function. Moreover, the first portion comprises less windowcoefficients than the second portion but an energy value of the windowcoefficients in the first portion is higher than an energy value of thewindow coefficients of the second portion. The first group of windowcoefficients is used for windowing later time-domain samples and thesecond group of window coefficients is used for windowed earliertime-domain samples. This form of the window function provides theopportunity of processing time-domain samples with window coefficientshaving higher energy values earlier. This is a result of the describeddistribution of window coefficients to the two portions and theirapplication to the sequence of time-domain audio samples. As aconsequence, employing such a window function can reduce the delayintroduced by the filterbank at a constant quality level or enables animproved quality level based on a constant delay level.

Accordingly, in the case of an embodiment of the present invention inthe form of an apparatus for generating time-domain audio samples and acorresponding method, a synthesis windower may use a synthesis windowfunction, which comprises a sequence of window coefficients orderedcorrespondingly in a first (consecutive) portion and (consecutive)second portion. Also in the case of a synthesis window function, anenergy value or an overall energy value of a window coefficient in thefirst portion is higher than an energy value or an overall energy valueof a window coefficient of a second portion, wherein the first portioncomprises less window coefficients than the second portion. Due to thisdistribution of the window coefficients among the two portions and thefact that the synthesis windower uses the first portion of windowercoefficients for windowing later time-domain samples and the secondportion of window coefficients for windowing earlier time-domainsamples, the previously described effects and advantages also apply to asynthesis filterbank or a corresponding embodiment of a method.

Detailed descriptions of synthesis window functions and analysis windowfunctions employed in the framework of some embodiments of the presentinvention will be described later in more detail. In many embodiments ofthe present invention, the sequence of window coefficients of thesynthesis window function and/or of the analysis window functioncomprise exactly the first group and the second group of windowcoefficients. Moreover, each of the window coefficients of the sequenceof window coefficients belongs exactly to one of the first group and thesecond group of window coefficients.

Each of the two groups comprises exactly one portion of the sequence ofwindow coefficients in a consecutive manner. In the present description,a portion comprises a consecutive set of window coefficients accordingto the sequence of the window coefficients. In embodiments according tothe present invention, each of the two groups (first and second group)comprises exactly one portion of the sequence of the window coefficientsin the above-explained manner. The respective groups of windowcoefficients do not comprise any window coefficients, which do notbelong to the exactly one portion of the respective group. In otherword, in many embodiments of the present invention, each of the firstand the second group of window coefficients comprises only the firstportion and the second portion of window coefficients without comprisingfurther window coefficients.

In the framework of the present description, a consecutive portion ofthe sequence of window coefficients is to be understood as a connectedset of window coefficients in the mathematical sense, wherein the setdoes not lack window coefficients compared to the sequence of windowcoefficients, which would be lying in a range (e.g. index range) of thewindow coefficients of the respective portion. As a consequence, in manyembodiments of the present invention, the sequence of windowcoefficients is divided exactly into two connected portions of windowcoefficients, which form each one of the first or the second groups ofwindow coefficients. In these cases, each window coefficient comprisedin the first group of window coefficients is either arranged before orafter each of the window coefficients of the second group of windowcoefficients with respect to the overall sequence of the windowcoefficients.

In yet other words, in many embodiments according to the presentinvention the sequence of window coefficients is divided exactly intotwo groups or portions without leaving any window coefficients out.According to the sequence of the window coefficients, which alsorepresents also an order of these, each of the two groups or portionscomprise all window coefficients up to (but excluding) or beginning from(including) a border window coefficient. As an example, the firstportion or first group may comprise window coefficients having indicesfrom 0 to 95 and from 96 to 639 in the case of a window functioncomprising 640 window coefficients (having indices of 0 to 639). Here,the border window coefficient would be that corresponding to index 96.Naturally, other examples are also possible (e.g. 0 to 543 and 544 to639).

The detailed exampled implementation of an analysis filterbank describedin the following provides a filter length covering 10 blocks of inputsamples while causing a system delay of only 2 blocks, which is thecorresponding delay as introduced by a MDCT (modified discrete cosinetransform) or a MDST (modified discrete sine transform). One differenceis due to the longer filter length covering 10 blocks of input samplescompared to an implementation of a MDCT or MDST that the overlap isincreased from 1 block in the case of MDCT and MDST to an overlap of 9blocks. However, further implementations can also be realized covering adifferent number of blocks of input samples, which are also referred toas audio input samples. Moreover, other trade-offs can also beconsidered and implemented.

FIG. 1 shows a block diagram of an analysis filterbank 100 as anembodiment of an apparatus for generating audio subband values in audiosubband channels. The analysis filterbank 100 comprises an analysiswindower 110 for windowing a frame 120 of time-domain audio inputsamples. The frame 120 comprises T blocks 130-1, . . . , 130-T blocks oftime-domain audio (input) samples, wherein T is a positive integer andequal to 10 in the case of the embodiment shown in FIG. 1. However, theframe 120 may also comprise a different number of blocks 130.

Both, the frame 120 and each of the blocks 130 comprises time-domainaudio input samples in a time sequence extending from an early sample toa later sample according to a time line as indicated by an arrow 140 inFIG. 1. In other words, in the illustration as shown in FIG. 1, thefurther the time-domain audio sample, which in this case is also atime-domain audio input sample, is to the right, the later thecorresponding time-domain audio sample is with respect to the sequenceof the time-domain audio sample.

The analysis windower 110 generates based on the sequence of time-domainaudio samples windowed samples in the time-domain, which are arranged ina frame 150 of windowed samples. According to the frame 120 oftime-domain audio input samples, also the frame of windowed samples 150comprises T blocks of windowed samples 160-1, . . . , 160-T. Inembodiments of the present invention each of the blocks of windowedsamples 160 comprises the same number of windowed samples as the numberof time-domain audio input samples of each block 130 of time-domainaudio input samples. Hence, when each of the blocks 130 comprises Ntime-domain input audio samples, the frame 120 and the frame 150 eachcomprise T·N samples. In this case, N is a positive integer, which may,for instance, acquire the values of 32 or 64. For T=10, the frames 120,150 each comprise 320 and 640, respectively, in the case above.

The analysis window 110 is coupled to a calculator 170 for calculatingthe audio subband values based on the windowed samples provided by theanalysis windower 110. The audio subband values are provided by thecalculator 170 as a block 180 of audio subband values, wherein each ofthe audio subband values corresponds to one audio subband channel. In anembodiment, also the block 180 of audio subband values comprises Nsubband values.

Each of the audio subband channels corresponds to a characteristiccenter frequency. The center frequencies of the different audio subbandchannels may, for instance, be equally distributed or equally spacedwith respect to the frequency bandwidth of the corresponding audiosignal as described by the time-domain audio input samples provided tothe analysis filterbank 100.

The analysis windower 110 is adapted to windowing the time-domain audioinput samples of the frame 120 based on an analysis window functioncomprising a sequence of window coefficients having a first number ofwindow coefficients to obtain the windowed samples of the frame 150. Theanalysis window 110 is adapted to performing the windowing of the frameof time-domain audio samples 120 by multiplying the values of thetime-domain audio samples with the window coefficients of the analysiswindow function. In other words, the windowing comprises andelement-wise multiplying of the time-domain audio samples with acorresponding window coefficient. As both, the frame 120 of time-domainaudio samples and the window coefficients comprise a correspondingsequence, the element-wise multiplication of the window coefficients andthe time-domain audio samples is carried out according to the respectivesequences, for instance, as indicated by a sample and window coefficientindex.

In embodiments of the present invention, the window functions used forwindowing the frame of time-domain audio input samples is generatedbased on a larger window function comprising a larger second number ofwindow coefficients by employing an interpolation scheme as, forinstance, outlined in the context of FIGS. 3 and 4. The larger windowfunction typically comprises an even number of window coefficients andmay, for instance, be asymmetric with respect to the sequence of windowcoefficients. Also symmetric window functions may be employed.

The window function 190 used for windowing the frame 120 of time-domaininput samples is, for instance, obtained by the analysis windower 110 orthe filterbank 100 interpolating the window coefficients of the largerwindow function. In embodiments according to the present invention, thisis carried out, for instance, by interpolating consecutive windowcoefficients of the larger window function. Here a linear, a polynomialor a spline-based interpolation scheme may be employed.

When, for instance, each window coefficient of the larger windowfunction is used once to generate a window coefficient of the windowfunction and the second number is even, the number of windowcoefficients of the window function 190 (first number) is half thesecond number. Such an interpolation may be based on a linearinterpolation, an example of which will be outlined in the context ofequation (15) later. However, also other interpolation schemes may beemployed as outlined.

In embodiments of the present invention in the form of an analysisfilterbank 100 as shown in FIG. 1, the analysis window function, as wellas the synthesis window function in the case of a synthesis filterbank,may for instance comprise real-valued windowed coefficients only. Inother words, each of the window coefficients attributed to a windowcoefficient index is a real value.

The window coefficients together form the respective window function, anexample of which is shown in FIG. 1 as an analysis window function 190.In the following, window functions will be considered, which allow areduction of the delay when used in the context of the describedfilterbanks. However, embodiments of the present invention are notlimited to such low-delay window functions.

The sequence of window coefficients forming the analysis window function190 comprises a first group 200 and a second group 210 of windowcoefficients. The first group 200 comprises a first consecutive andconnected portion of the window coefficients of the sequence of windowcoefficients, whereas the second group 210 comprises a consecutive andconnected second portion of a window coefficient. Together with thefirst portion in the first group 200, they form the whole sequence ofwindow coefficients of the analysis window function 190. Moreover, eachwindow coefficient of the sequence of window coefficients belongs eitherto the first portion or the second portion of window coefficients sothat the whole analysis window function 190 is made up by the windowcoefficient of the first portion and the second portion. The firstportion of window coefficients is, hence, identical to the first group200 of window coefficients and the second portion is identical to thesecond group 210 of window coefficients as indicated by thecorresponding arrows 200, 210 in FIG. 1.

The number of window coefficients in the first group 200 of the firstportion of window coefficients is smaller than the number of windowcoefficients in the second group of the second portion of windowcoefficients. However, an energy value or a total energy value of thewindow coefficients in the first group 200 is higher than an energyvalue or total energy value of the window coefficients in the secondgroup 210. As will be outlined later, an energy value of a set of windowcoefficients is based on a sum of the squares of the absolute values ofthe corresponding window coefficients.

In embodiments according to the present invention, the analysis windowfunction 190 as well as a corresponding synthesis window function may betherefore asymmetric with respect to the sequence of window coefficientsor an index of a window coefficient. Based on a definition set of windowcoefficient indices over which the analysis window function 190 isdefined, the analysis window function 190 is asymmetric, when for allreal numbers n a further real number n₀ exists so that the absolutevalue of window coefficient corresponding to the window coefficient ofthe window coefficient index (n₀−n) is not equal to the absolute valueof the window coefficient corresponding to the window coefficient index(n₀+n), when (n₀−n) and (n₀+n) belong to the definition set.

Moreover, as also schematically shown in FIG. 1, the analysis windowfunction 190 comprises sign changes at which the product of twoconsecutive window coefficients is negative. More details and furtherfeatures of possible window functions according to embodiments of thepresent invention will be discussed in more detail in the context ofFIGS. 11 to 19.

As indicated earlier, the frame of windowed samples 150 comprises asimilar block structure with individual blocks 160-1, . . . , 160-T asthe frame 120 of individual time-domain input samples. As the analysiswindower 110 is adapted to windowing the time-domain audio input samplesby multiplying these values with the window coefficients of the analysiswindow function 190, the frame 150 of windowed samples is also in thetime-domain. The calculator 170 calculates the audio subband values, orto be more precise, the block 180 of audio subband values using theframe 150 of windowed samples and performs a transfer from thetime-domain into the frequency-domain. The calculator 170 can thereforebe considered to be a time/frequency converter, which is capable ofproviding the block 180 of audio subband values as a spectralrepresentation of the frame 150 of windowed samples.

Each audio subband value of the block 180 corresponds to one subbandhaving a characteristic frequency. The number of audio subband valuescomprised in the block 180 is also sometimes referred to as a bandnumber.

In many embodiments according to the present invention, the number ofaudio subband values in block 180 is identical to the number oftime-domain audio input samples of each of the blocks 130 of the frame120. In the case that the frame 150 of windowed samples comprises thesame block-wise structure as the frame 120 so that each of the blocks160 of windowed samples also comprise the same number of windowedsamples as the block of these time-domain audio input samples 130, theblock 180 of audio subband values naturally also comprises the samenumber as the block 160.

The frame 120 can optionally be generated, based on a block of freshtime-domain audio input samples 220 by shifting the blocks 130-1, . . ., 130-(T−1) by one block in the opposite direction of the arrow 140indicating the time-direction. Thereby, a frame 120 of time-domain audioinput samples to be processed is generated by shifting the (T−1) latestblocks of a directly preceding frame 120 of time-domain audio samples byone block towards the earlier time-domain audio samples and by addingthe fresh block 220 of fresh time-domain audio samples as the new block130-1 comprising the latest time-domain audio samples of the presentframe 120. In FIG. 1 this is also indicated by a series of dashed arrows230 indicating shifting the blocks 130-1, . . . , 130-(T−1) in theopposite direction of the arrow 140.

Due to this shifting of the blocks 130 in the opposite direction of thetime as indicated by arrow 140, a present frame 120 to be processed,comprises the block 130-(T−1) of the directly preceding frame 120 as thenew block 130-T. Accordingly, the blocks 130-(T−1), . . . , 130-2 of thepresent frame 120 to be processed are equal to the block 130-(T−2), . .. , 130-1 of the directly preceding frame 120. The block 130-T of thedirectly preceding frame 120 is discarded.

As a consequence, each time-domain audio sample of the fresh block 220will be processed T-times in the framework of T consecutive processingsof T consecutive frames 120 of time-domain audio input samples. Hence,each time-domain audio input sample of the fresh block 220 contributes,not only to T different frames 120, but also to T different frames 150of windowed samples and T blocks 180 of audio subband values. Asindicated before, in an embodiment according to the present invention,the number of blocks T in the frame 120 is equal to 10, so that eachtime-domain audio sample provided to the analysis filterbank 100contributes to 10 different blocks 180 of audio subband values.

In the beginning, before a single frame 120 is processed by the analysisfilterbank 100, the frame 120 can be initialized to a small absolutevalue (below a predetermined threshold), for instance the value 0. Aswill be explained in more detail below, the shape of the analysis windowfunction 190 comprises a center point or a “center of mass”, whichtypically corresponds to or lies between two window coefficient indicesof the first group 200.

As a consequence, the number of fresh blocks 220 to be inserted into theframe 120 is small, before the frame 120 is filled at least to a pointso that portions of the frame 120 are occupied by non-vanishing (i.e.non-zero-valued) values which correspond to window coefficients having asignificant contribution in terms of their energy values. Typically, thenumber of blocks to be inserted into frame 120 before a “meaningful”processing can begin, is 2 to 4 blocks depending on the shape of theanalysis window function 190. Hence, the analysis filterbank 100 iscapable of providing blocks 180 faster than a corresponding filterbankemploying, for instance, a symmetric window function. As typically thefresh blocks 220 are provided to the analysis filterbank 100 as a whole,each of the fresh blocks corresponds to a recording or sampling time,which is essentially given by the length of the block 220 (i.e. thenumber of time-domain audio input samples comprised in block 220) andthe sampling rate or sampling frequency. Therefore, analysis windowfunction 190, as incorporated into an embodiment of the presentinvention, leads to a reduced delay before the first and the followingblocks 180 of audio subband values can be provided or output by thefilterbank 100.

As a further option, the apparatus 100 can be capable of generating asignal or incorporating a piece of information concerning the analysiswindow function 190 used in generating the frame 180 or concerning asynthesis window function to be used in the framework of a synthesisfilterbank. Thus, the analysis filter function 190 can, for instance, bea time- or index-reversed version of the synthesis window function to beused by the synthesis filterbank.

FIG. 2 a shows a block diagram of an embodiment of an apparatus 300 forgenerating time-domain audio samples based on the block of audio subbandvalues. As previously explained, an embodiment of the present inventionin the form of an apparatus 300 for generating time-domain audio samplesis often also referred to as a synthesis filterbank 300 as the apparatusis capable of generating time-domain audio samples, which can inprinciple be played back, based on audio subband values which comprisespectral information concerning an audio signal. Hence, the synthesisfilterbank 300 is capable of synthesizing time-domain audio samplesbased on audio subband values, which can for instance be generated by acorresponding analysis filterbank 100.

FIG. 2 a shows a block diagram of the synthesis filterbank 300comprising a calculator 310 to which a block 320 of audio subband values(in the frequency-domain) is provided. The calculator 310 is capable ofcalculating a frame 330 comprising a sequence of intermediatetime-domain samples from the audio subband values of the block 320. Theframe 330 of intermediate time-domain samples comprises in manyembodiments according to the present invention also a similar blockstructure as for instance the frame 150 of windowed samples of theanalysis filterbank 100 of FIG. 1. In these cases, the frame 330comprises blocks 340-1, . . . , 340-T blocks of intermediate time-domainsamples.

The sequence of intermediate time-domain samples of the frame 330, aswell as each block 340 of intermediate time-domain samples comprise anorder according to the time as indicated by an arrow 350 in FIG. 2 a. Asa consequence, the frame 330 comprises an early intermediate time-domainsample in block 340-T and a latest intermediate time-domain sample inblock 340-1, which represent the first and the last intermediatetime-domain sample for the frame 330, respectively. Also each of theblocks 340 comprises a similar order. As a consequence, in embodimentsof a synthesis filterbank the terms “frame” and “sequence” can often beused synonymously.

The calculator 310 is coupled to a synthesis windower 360 to which theframe 330 of intermediate time-domain samples is provided. The synthesiswindower is adapted to windowing the sequence of intermediatetime-domain samples using a synthesis window function 370 schematicallydepicted in FIG. 2 a. As an output, the synthesis windower 360 providesa frame 380 of windowed intermediate time-domain samples, which may alsocomprise a block-wise structure of blocks 390-1, . . . , 390-T.

The frames 330 and 380 may comprise T blocks 340 and 390, respectively,wherein T is a positive integer. In an embodiment according to thepresent invention in the form of a synthesis filterbank 300, the numberof blocks T is equal to 10. However, in different embodiments, alsodifferent numbers of blocks may be comprised in one of the frames. To bemore precise, in principle the number of blocks T may be larger or equalto 3, or larger than or equal to 4, depending on the circumstances ofthe implementation and the previously explained trade-offs forembodiments according to the present invention comprising a block-wisestructure of frames for both a synthesis filterbank 100 and a synthesisfilterbank 300.

The synthesis windower 360 is coupled to an overlap-adder output stage400, to which the frame 380 of windowed intermediate time-domain samplesis provided. The overlap-adder output stage 400 is capable of processingthe windowed intermediate time-domain samples to obtain a block 410 oftime-domain samples. The block 410 of the time-domain (output) samplescan then for instance be provided to further components for furtherprocessing, storing or transforming into audible audio signals.

The calculator 310 for calculating the sequence of time-domain samplescomprised in the frame 330 is capable of transferring data from thefrequency-domain into the time-domain. Therefore, the calculator 310 maycomprise a frequency/time converter capable of generating a time-domainsignal of the spectral representation comprised in the block 320 ofaudio subband values. As was explained in the context of the calculator170 of the analysis filterbank 100 shown in FIG. 1, each of the audiosubband values of the block 320 corresponds to an audio subband channelhaving a characteristic center frequency.

In contrast to this, the intermediate time-domain samples comprised inthe frame 330 represent in principle information in the time-domain. Thesynthesis windower 360 is capable and adapted to windowing the sequenceof intermediate time-domain samples comprised in the frame 330 using thesynthesis window function 370 as schematically depicted in FIG. 2 a.

As already outlined in the context of FIG. 1, the synthesis windower 360also uses a synthesis window function 370, which is obtained byinterpolation of a larger window function comprising a second number ofwindow coefficients. The second number is, hence, larger than a firstnumber of window coefficients of the synthesis window function 370 usedfor windowing the intermediate time-domain samples of the frame 330.

The synthesis window function 370 may, for instance, be obtained by thesynthesis windower 360 or the filterbank 300 (the apparatus) performingone of the previously outlined interpolation schemes. The windowcoefficients of the synthesis window function may, for instance, begenerated based on a linear, a polynomial or a spline-basedinterpolation. Moreover, in embodiments according to the presentinvention the interpolation may be based on using consecutive windowcoefficients of the larger window function. When each window coefficientof the larger window function is used exactly once, the window function370 comprising the (smaller) first number of window coefficients may forinstance comprise exactly half the number of window coefficients of thelarger window function, when the second number is even. In other words,in this case the second number may be twice the first number. However,also other interpolation scenarios and schemes may be implemented in theframework of embodiments of the present invention.

In the following, the case of a so-called low-delay window function willbe considered more closely. As indicated earlier, embodiments accordingto the present invention are by far not limited to these windowfunctions. Also other window functions, such as symmetric windowfunctions, may be used.

The synthesis window function 370 comprises a sequence of windowcoefficients, which also comprises a first group 420 and a second group430 of window coefficients as previously explained in the context of thewindow function 190 with a first group 200 and a second group 210 ofwindow coefficients.

The first group 420 of window coefficients of the synthesis windowfunction 370 comprises a first consecutive portion of the sequence ofwindow coefficients. Similarly, the second group 430 of coefficientsalso comprises a second consecutive portion of the sequence of windowcoefficients, wherein the first portion comprises less windowcoefficients than the second portion and wherein an energy value ortotal energy value of the window coefficients in the first portion ishigher than the corresponding energy value of the window coefficients ofthe second portion. Further features and properties of the synthesiswindow function 370 may be similar to the corresponding features andproperties of the analysis window function 190 as schematically depictedin FIG. 1. As a consequence, reference is hereby made to thecorresponding description in the framework of the analysis windowfunction 190 and the further description of the window functions withrespect to FIGS. 11 to 19, wherein the first group 200 corresponds tothe first group 420 and the second group 210 corresponds to the secondgroup 430.

For instance the portions comprised in the two groups 420, 430 of windowcoefficients typically each form a consecutive and connected set ofwindow coefficients together comprising all window coefficients of thesequence of window coefficients of the window function 370. In manyembodiments according to the present invention, the analysis windowfunction 190 as depicted in FIG. 1 and the synthesis window function 370as depicted in FIG. 2 a are based on each other. For instance, theanalysis window function 190 may be a time-reversed or index-reversedversion of the synthesis window function 370. However, also otherrelations between the two window functions 190, 370 may be possible. Itmay be advisable to employ a synthesis window function 370 in theframework of the synthesis windower 360, which is related to theanalysis window function 190, which was employed in the course ofgenerating (optionally before further modifications) of the block 320 ofaudio subband vales provided to the synthesis filterbank 300.

As outlined in the context of FIG. 1, the synthesis filterbank 300 inFIG. 2 a may optionally be adapted such that the incoming block 320 maycomprise additional signals or additional pieces of informationconcerning the window functions. As an example, the block 320 maycomprise information concerning the analysis window function 190 usedfor generating the block 320 or concerning the synthesis window function370 to be used by the synthesis windower 360. Hence, the filterbank 300may be adapted to isolating the respective information and to providethese to the synthesis windower 360.

The overlap-adder output stage 400 is capable of generating the block410 of time-domain samples by processing the windowed intermediatetime-domain samples comprised in the frame 380. In different embodimentsaccording to the present invention, the overlap-adder output stage 4000may comprise a memory for temporarily storing previously received frames380 of windowed intermediate time-domain samples. Depending onimplementational details, the overlap-adder output stage 400 may forinstance comprise T different storage positions comprised in the memoryfor storing an overall number of T frames 380 of windowed intermediatetime-domain samples. However, also a different number of storagepositions may be comprised in the overlap-adder output stage 400 asnecessitated. Moreover, in different embodiments according to thepresent invention, the overlap-adder output stage 400 may be capable ofproviding the block 410 of time-domain samples based on a single frame380 of intermediate time-domain samples alone. Embodiments of differentsynthesis filterbanks 300 will be explained in more detail later.

FIG. 2 b illustrates a functional principle according to an embodimentof the present invention in the form of a synthesis filterbank 300, inwhich the generation of the window function 370 by interpolation is notfocused on for the sake of simplicity only.

The block 320 of audio subband values is first transferred from thefrequency-domain into the time-domain by the calculator 310, which isillustrated in FIG. 2 b by an arrow 440. The resulting frame 320 ofintermediate time-domain samples comprising the blocks 340-1, . . . ,340-T of intermediate time-domain samples is then windowed by thesynthesis windower 360 (not shown in FIG. 2 b) by multiplying thesequence of intermediate time-domain samples of the frame 320 with thesequence of window coefficients of the synthesis window function 370 toobtain the frame 380 of windowed intermediate time-domain samples. Theframe 380 again comprises the blocks 390-1, . . . , 390-T of windowedintermediate time-domain samples, together forming the frame 380 ofwindowed intermediate time-domain samples.

In the embodiment shown in FIG. 2 b of an inventive synthesis filterbank300, the overlap-adder output stage 400 is then capable of generatingthe block 410 of time-domain output samples by adding for each indexvalue of the time-domain audio samples of the block 410, the windowedintermediate time-domain samples of one block 390 of different frames380. As illustrated in FIG. 2 b, the time-domain audio samples of theblock 410 are obtained by adding for each audio sample index onewindowed intermediate time-domain sample of block 390-1 of the frame380, processed by the synthesis windower 360 in the current round and aspreviously described, the corresponding intermediate time-domain sampleof the second block 390-2 of a frame 380-1 processed immediately beforeframe 380 and stored in a storage position in the overlap-adder outputstage 400. As illustrated in FIG. 2 b, further corresponding windowedintermediate time-domain samples of further blocks 390 (e.g. block 390-3of frame 380-2, block 390-4 of frame 380-3, block 390-5 of frame 380-4)processed by the synthesis filterbank 300 before may be used. The frames380-2, 380-3, 380-4 and optionally further frames 380 have beenprocessed by the synthesis filterbank 300 in previous rounds. The frame380-2 has been immediately processed before the frame 380-1 and,accordingly, frame 380-3 was immediately generated before frame 380-2and so on.

The overlap-adder output stage 400 as employed in the embodiment iscapable of summing up for each index of the block 410 of time-domain(output) samples T different blocks 390-1, . . . , 390-T of windowedintermediate time-domain samples from T different frames 380, 380-1,380-(T−1). Hence, apart from the first T blocks processed, each of thetime-domain (output) samples of the block 410 is based on T differentblocks 320 of audio subband values.

As in the case of the embodiment of the present invention an analysisfilterbank 100 described in FIG. 1, due to the form of the synthesiswindow function 370, the synthesis filterbank 300 offers the possibilityof quickly providing the block 410 of time-domain (output) samples. Thisis also a consequence of the form of the window function 370. As Thefirst group 420 of window coefficients correspond to a higher energyvalue and comprise less window coefficients than the second group 430,the synthesis windower 360 is capable of providing “meaningful” frames380 of windowed samples when the frame 330 of intermediate time-domainsamples is filled so that at least the window coefficients of the firstgroup 420 contribute to the frame 380. The window coefficients of thesecond group 430 exhibit a smaller contribution due to their smallerenergy value.

Therefore, when at the beginning, the synthesis filterbank 300 isinitialized with 0, the provision of blocks 410 can in principle, bestarted when only a few blocks 320 of audio subband values have beenreceived by the synthesis filterbank 300. Therefore, also the synthesisfilterbank 300 enables significant delay reduction compared to thesynthesis filterbank having for instance a symmetric synthesis windowfunction.

As indicated earlier, the calculators 170 and 310 of the embodimentsshown in FIGS. 1 and 2 a can be implemented as real-valued calculatorsgenerating or being capable of processing real-valued audio subbandvalues of the blocks 180 and 320, respectively. In these cases, thecalculators may for instance be implemented as real-valued calculatorsbased on harmonic oscillating functions such as the sine-function or thecosine-function. However, also complex-valued calculators can beimplemented as the calculators 170, 310. In these cases the calculatorsmay for instance be implemented on the basis of complexexponential-functions or other harmonic complex-valued functions. Thefrequency of the real-valued or complex-valued oscillations usuallydepends on the index of the audio subband value, which is sometimes alsoreferred to as the band index or the subband index of the specificsubband. Moreover, the frequency may be identical or depend on thecenter frequency of the corresponding subband. For instance, thefrequency of the oscillation may be multiplied by a constant factor,shifted with respect to the center frequency of the correspondingsubband or may be depending on a combination of both modifications.

A complex-valued calculator 170, 310 may be constructed or implementedbased on real-value calculators. For instance, for a complex-valuedcalculator an efficient implementation can in principle be used forboth, the cosine- and the sine-modulated part of a filterbankrepresenting the real and the imaginary part of a complex-valuedcomponent. This means that it is possible to implement both, thecosine-modulated part and the sine-modulated part based on, forinstance, the modified DCT-IV- and DST-IV-structures.

Moreover, further implementations might employ the use of a FFT(FFT=Fast Fourier Transform) optionally being implemented jointly forboth, the real part and the part of the complex-modulated calculatorsusing one FFT or instead using one separate FFT stage for eachtransform.

Mathematical Description

The following sections will describe an example of the embodiments of ananalysis filterbank and the synthesis filterbank with multiple overlapsof 8 blocks to the part, which do not cause further delay, as explainedabove, and one block to the future, which causes the same delay as for aMDCT/MDST-structure (MDCT=Modified Discrete Cosine Transform;MDST=Modified Discrete Sine Transform). In other words, in the followingexample, the parameter T is equal to 10.

First, a description of a complex-modulated low-delay analysisfilterbank will be given. As illustrated in FIG. 1, the analysisfilterbank 100 comprises the transformation steps of an analysiswindowing performed by the analysis windower 110 and an analysismodulation performed by the calculator 170. The analysis windowing isbased on the equationz _(i,n) =w(10N−1−n)·x _(i,n) for 0≦n≦10·N,  (1)wherein, z_(i,n) is the (real-valued) windowed sample corresponding tothe block index i and the sample index n of the frame 150 shown inFIG. 1. The value x_(i,n) is the (real-valued) time input samplecorresponding to the same block index i and sample index n. The analysiswindow function 190 is represented in equation (1) by its real-valuedwindow coefficients w(n), wherein n is also the window coefficient indexin the range indicated in equation (1). As already previously explained,the parameter N is the number of samples in one block 220, 130, 160,180.

From the arguments of the analysis window function w(10N−1−n) can beseen that the analysis window function represents a flipped version or atime-reversed version of the synthesis window function, which isactually represented by the window coefficient w(n).

The analysis modulation carried out by the calculator 170 in theembodiment shown in FIG. 1, is based on the two equations

$\begin{matrix}{{X_{{{Re}\mspace{14mu}{al}},i,k} = {2 \cdot {\sum\limits_{n = {{- 8}N}}^{{2N} - 1}{z_{i,n}{\cos\left( {\frac{\pi}{N}\left( {n + n_{0}} \right)\left( {k + \frac{1}{2}} \right)} \right)}}}}}{and}} & (2) \\{X_{{{Im}\mspace{11mu}{ag}},i,k} = {2 \cdot {\sum\limits_{n = {{- 8}N}}^{{2N} - 1}{z_{i,n}{\sin\left( {\frac{\pi}{N}\left( {n + n_{0}} \right)\left( {k + \frac{1}{2}} \right)} \right)}}}}} & (3)\end{matrix}$for the spectral coefficient index or band index k being an integer inthe range of0≦k≦N.  (4)

The values X_(Real,i,k) and X_(imag,i,k) represent the real part and theimaginary part of the complex-valued audio subband value correspondingto the block index i and the spectral coefficient index k of block 180.The parameter n₀ represents an index option, which is equal ton ₀ =−N/2+0.5.  (5)

The corresponding complex-modulated low-delay synthesis filterbankcomprises the transformation steps of a synthesis modulation, asynthesis windowing and an overlap-add as will be described.

The synthesis modulation is based on the equation

$\begin{matrix}{{x_{i,n}^{\prime} = {\frac{1}{N} \cdot \begin{bmatrix}{{\sum\limits_{k = 0}^{N - 1}{X_{{Real},i,k}\cos\left( {\frac{\pi}{N}\left( {n + n_{0}} \right)\left( {k + \frac{1}{2}} \right)} \right)}} +} \\{\sum\limits_{k = 0}^{N - 1}{X_{{Imag},i,k}{\sin\left( {\frac{\pi}{N}\left( {n + n_{0}} \right)\left( {k + \frac{1}{2}} \right)} \right)}}}\end{bmatrix}}}\mspace{14mu}} & (6)\end{matrix}$wherein x′_(i,n) is an intermediate time-domain sample of the frame 330corresponding to the sample index n and the block index i. Once againthe parameter N is an integer indicating the length of the block 320,340, 390, 410, which is also referred to as transform block length or,due to the block-wise structure of the frames 330, 380, as an offset tothe previous block. Also the further variables and parameters have beenintroduced above, such as the spectral coefficient index k and theoffset n₀.

The synthesis windowing carried out by the synthesis windower 360 in theembodiment shown in FIG. 2 a is based on the equationz′ _(i,n) =w(n)·x′ _(i,n) for 0≦n≦10·N,  (7)wherein z′_(i,n) is the value of the windowed intermediate time-domainsample corresponding to the sample index n and the block index i of theframe 380.

The transformation stamp of the overlap-add is based on the equationout_(i,n) =z′ _(i,n) +z′ _(i-1,n+N) +z′ _(i-2,n+2N) +z′ _(i-3,n+3N) +z′_(i-4,n+4N) +z′ _(i-5,n+5N) +z′ _(i-6,n+6N) +z′ _(i-7,n+7N) +z′_(i-8,n+8N) +z′ _(i-9,n+9N),  (8)for 0≦n<Nwherein out_(i,n) represents the time-domain (output) samplecorresponding to the sample index n and the block index i. Equation (8),hence illustrates the overlap-add operation as carried out theoverlap-adder output stage 400 as illustrated in the lower part of FIG.2 b.

However, embodiments according to the present invention are not limitedto complex-modulated low-delay filterbanks allowing for an audio signalprocessing with one of these filterbanks. A real-valued implementationof a low-delay filterbank for an enhanced low-delay audio coding canalso be implemented. As a comparison, for instance, equations (2) and(6) in terms of a cosine-part reveals, the cosine-contribution of theanalysis modulation and the synthesis modulation show a comparablestructure when considering that of a MDCT. Although the design method inprinciple allows an extension of the MDCT in both directions concerningtime, only an extension of E (=T−2) blocks to the past is applied here,where each of the T blocks comprises N samples. The frequencycoefficient X_(i,k) of band k and block i inside an N-channel or N-bandanalysis filterbank can be summarized by

$\begin{matrix}{X_{i,k}^{\prime} = {{- 2}{\sum\limits_{n = {{- E} \cdot N}}^{{2N} - 1}{{w_{a}(n)} \cdot {x(n)} \cdot {\cos\left( {\frac{\pi}{N}\left( {n + \frac{1}{2} - \frac{N}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right)}}}}} & (9)\end{matrix}$for the spectral coefficient index k as defined by equation (4). Here,once again n is a sample index and w_(a) is the analysis windowfunction.

For the sake of completeness, the previously given mathematicaldescription of the complex-modulated low-delay analysis filterbank canbe given in the same summarizing form as equation (9) by exchanging thecosine-function with the complex-valued exponential-function. To be moreprecise, with the definition and variables given above, the equations(1), (2), (3) and (5) can be summarized and extended according to

$\begin{matrix}{{X_{i,k}^{\prime} = {{- 2}{\sum\limits_{n = {{- E} \cdot N}}^{{2N} - 1}{{w_{a}(n)} \cdot {x(n)} \cdot {\exp\left( {{j \cdot \frac{\pi}{N}}\left( {n + \frac{1}{2} - \frac{N}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right)}}}}},} & (10)\end{matrix}$wherein in contrast to the equations (2) and (3), the extension of 8blocks into the past has been replaced by the variable E(=8).

The steps of the synthesis modulation and the synthesis windowing, asdescribed for the complex case in equations (6) and (7), can besummarized in the case of a real-valued synthesis filterbank. The frame380 of windowed intermediate time-domain samples, which is also referredto as the demodulated vector, is given by

$\begin{matrix}{z_{i,n}^{\prime} = {{- \frac{1}{N}}{\sum\limits_{k = 0}^{N - 1}{{{w_{s}(n)} \cdot X_{i,k}}{\cos\left( {\left( {\frac{\pi}{N}\left( {n + \frac{1}{2} - \frac{N}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right),} \right.}}}}} & (11)\end{matrix}$wherein z′_(i,n) is the windowed intermediate time-domain samplecorresponding to the band index i and the sample index n. The sampleindex n is once again an integer in the range of0≦n≦N(2+E)=N·T  (12)and w_(s)(n) is the synthesis window, which is compatible with theanalysis window w_(a)(n) of equation (9).

The transformation step of the overlap-add is then given by

$\begin{matrix}{x_{i,n}^{\prime} = {\sum\limits_{l = {({E + 1})}}^{0}z_{{i + l},{n - {l \cdot N}}}^{\prime}}} & (13)\end{matrix}$wherein x′_(i,n) is the reconstructed signal, or rather a time-domainsample of the block 410 as provided by the overlap-add output stage 400shown in FIG. 2 a.

For the complex-valued synthesis filterbank 300, the equations (6) and(7) can be summarized and generalized with respect to the extension ofE(=8) blocks to the path according to

$\begin{matrix}{z_{i,n}^{\prime} = {{- \frac{1}{N}}{\sum\limits_{k = 0}^{N - 1}{{w_{s}(n)} \cdot {{Re}\left( {X_{i,k}{\exp\left( {{{- j} \cdot \left( {\frac{\pi}{N}\left( {n + \frac{1}{2} - \frac{N}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right)},} \right.}} \right.}}}}} & (14)\end{matrix}$wherein j=√{square root over (−1)} is the imaginary unit. Equation (13)represents the generalized from of equation (8) and is also valid forthe complex-valued case.

As a direct comparison of equation (14) with equation (7) shows, thewindow function w(n) of equation (7) is the same synthesis windowfunction as w_(s)(n) of equation (14). As outlined before, the similarcomparison of equation (10) with the analysis window functioncoefficient w_(a)(n) with equation (1) shows that the analysis windowfunction is the time-reversed version of the synthesis window functionin the case of equation (1).

As both, an analysis filterbank 100 as shown in FIG. 1 and a synthesisfilterbank 300 as shown in FIG. 2 a offer a significant improvement interms of a trade-off between the delay on the one hand and the qualityof the audio process on the other hand, the filterbanks 100, 300 areoften referred to as low-delay filterbanks. The complex-valued versionthereof is sometimes referred to as complex-low-delay filterbank, whichis abbreviated by CLDFB. Under some circumstances, the term CLDFB is notonly used for the complex-valued version but also for the real-valuedversion of the filterbank.

As the previous discussion of the mathematical background has shown, theframework used for implementing the proposed low-delayed filterbanksutilizes a MDCT- or IMDCT-like (IMDCT=Inverse MDCT) structure, as knownfrom the MPEG-4 Standard, using an extended overlap. The additionaloverlap regions can be attached in a block-wise fashion to the left aswell as to the right side of the MDCT-like core. Here, only theextension to the right side (for the synthesis filterbank) is used,which works from past samples only and therefore does not cause anyfurther delay.

The inspection of the equations (1), (2) and (14) has shown that theprocessing is very similar to that of a MDCT or IMDCT. By only slightmodifications comprising a modified analysis window function andsynthesis window function, respectively, the MDCT or IMDCT is extendedto a modulated filterbank that is able to handle multiple overlaps andis very flexible concerning its delay. As for instance, equations (2)and (3) have shown the complex version is in principle obtained bysimply adding a sine-modulated to the given cosine-modulation.

Interpolation

As outlined in the context of FIGS. 1 and 2 a, both, the analysiswindower 110 and the synthesis windower 360 or the respectivefilterbanks 100, 300 are adapted to windowing the respective frames oftime-domain samples by multiplying each of the respective time-domainaudio samples with an individual window coefficient. Each of thetime-domain samples is, in other words, multiplied by an (individual)window coefficient, as for instance equations (1), (7), (9), (10), (11),and (14) have demonstrated. As a consequence, the number of windowcoefficients of the respective window function is typically identical tothe number of respective time-domain audio samples.

However, under certain implementational circumstances, it may beadvisable to implement a window function having a larger second numberof window coefficients compared to the actual window function having asmaller first number of coefficients, which is actually used during thewindowing of the respective frame or sequence of time-domain audiosamples. This may for instance be advisable in the case when memoryrequirements of a specific implementation may be more valuable thancomputational efficiency. A further scenario in which a downsampling ofthe window coefficients might become useful is in the case of theso-called dual rate approach, which is for instance employed in theframework of SBR systems (SBR=Spectral Band Replication). The concept ofSBR will be explained in more detail in the context of FIGS. 5 and 6.

In such a case, the analysis windower 110 or the synthesis windower 360may be further adapted such that the respective window function used forwindowing the time-domain audio samples provided to the respectivewindower 110, 360 is derived by an interpolation of window coefficientsof the larger window function having a larger second number of windowcoefficients.

The interpolation can for instance be carried out by a linear,polynomial or spline-based interpolation. For instance, in the case ofthe linear interpolation, but also in the case of a polynomial orspline-based interpolation, the respective windower 100, 360 may then becapable of interpolating the window coefficients of the window functionused for windowing based on two consecutive window coefficients of thelarger window function according to a sequence of the windowcoefficients of the larger window function to obtain one windowcoefficient of the window function.

Especially in the case of an even number of time-domain audio samplesand window coefficients, an implementation of an interpolation aspreviously described, results in a significant improvement of the audioquality. For instance, in the case of an even number N·T of time-domainaudio samples in one of the frames 120, 330, not using an interpolation,for instance, a linear interpolation, will result in severe aliasingeffects during the further processing of the respective time-domainaudio samples.

FIG. 3 illustrates an example of a linear interpolation based on awindow function (an analysis window function or a synthesis windowfunction) to be employed in context with frames comprising N·T/2time-domain audio samples. Due to memory restraints or otherimplementational details, the window coefficients of the window functionitself are not stored in a memory, but a larger window functioncomprising N T window coefficients are stored during appropriate memoryor are available otherwise. FIG. 3 illustrates in the upper graph, thecorresponding window coefficients c(n) as a function of the windowcoefficient indices n in the range between 0 and N·T−1.

Based on a linear interpolation of two consecutive window coefficientsof the window function having the larger number of window coefficients,as depicted in the upper graph of FIG. 3, an interpolated windowfunction is calculated based on the equation

$\begin{matrix}{{{ci}\lbrack n\rbrack} = {{\frac{1}{2}\left( {{c\left\lbrack {2n} \right\rbrack} + {c\left\lbrack {{2n} + 1} \right\rbrack}} \right)\mspace{14mu}{for}\mspace{14mu} 0} \leq n < {N \cdot {T/2.}}}} & (15)\end{matrix}$The number of interpolated window coefficients ci(n) of the windowfunction to be applied to a frame having N·T/2 time-domain audio samplescomprise half the number of window coefficients.

To illustrate this further, in FIG. 3 window coefficients 450-0, . . . ,450-7 are shown in the upper part of FIG. 3 corresponding to a windowcoefficient c(0), . . . , c(7). Based on these window coefficients andthe further window coefficients of the window function, an applicationof equation (15) leads to the window coefficients ci(n) of theinterpolated window function depicted in the lower part of FIG. 3. Forinstance, based on the window coefficients 450-2 and 450-3, the windowcoefficient 460-1 is generated based on equation (15), as illustrated bythe arrows 470 in FIG. 3. Accordingly, the window coefficient 460-2 ofthe interpolated window function is calculated based on the windowcoefficient 450-4, 450-5 of the window function depicted in the upperpart of FIG. 3. FIG. 3 shows the generation of further windowcoefficients ci(n).

To illustrate the aliasing cancellation achievable by the interpolateddownsampling of the window function, FIG. 4 illustrates theinterpolating of the window coefficients in the case of a sine windowfunction, which can, for instance, be employed in a MDCT. For the sakeof simplicity, the left half of the window function and the right halfof the window function are drawn over each other. FIG. 4 shows asimplified version of a sine window, comprising only 2·4 windowcoefficients or points for a MDCT having a length of 8 samples.

FIG. 4 shows four window coefficients 480-1, 480-2, 480-3 and 480-4 ofthe first half of the sine window and four window coefficients 490-1,490-2, 490-3 and 390-4 of the second half of the sine window. The windowcoefficient 490-1, . . . , 490-4 corresponds to the window coefficientindices 5, . . . , 8. The window coefficients 490-1, . . . , 490-4correspond to the second half of the length of the window function suchthat to the indices given N′=4 is to be added to obtain the realindices.

To reduce or even to achieve the cancellation of the aliasing effects asdescribed before, the window coefficient should fulfill the conditionw(n)·(N′−1−n)=w(N′+n)·w(2N′−1−n)  (16)as good as possible. The better relation (16) is fulfilled, the betterthe alias suppression or alias cancellation is.

Assuming the situation that a new window function having half the numberof window coefficients is to be determined for the left half of thewindow function, the following problem arises. Due to the fact that thewindow function comprises an even number of window coefficients (evennumbered downsampling), without employing an interpolation scheme asoutlined in FIG. 3, the window coefficients 480-1 and 480-3 or 480-2 and480-4 correspond to only one aliasing value of the original windowfunction or original filter.

This leads to an unbalanced proportion of spectral energy and leads toan unsymmetrical redistribution of the center point (center of mass) ofthe corresponding window function. Based on the interpolation equation(15) for the window coefficient w(n) of FIG. 4, the interpolated valuesI₁ and I₂ fulfill the aliasing relation (16) far better, and will hencelead to a significant improvement concerning the quality of theprocessed audio data.

However, by employing an even more elaborate interpolation scheme, forinstance a spline-based or another similar interpolation scheme, mighteven result in window coefficients, which fulfill the relation (16) evenbetter. A linear interpolation is in most cases sufficient and enables afast and efficient implementation.

The situation in the case of a typical SBR system employing a SBR-QMFfilterbank (QMF=Quadrature Mirror Filter), a linear interpolation oranother interpolation scheme does not need to be implemented as theSBR-QMF prototype filter comprises an odd number of prototype filtercoefficients. This means that the SBR-QMF prototype filter comprises amaximum value with respect to which the downsampling can be implementedso that the symmetry of the SBR-QMF prototype filter remains intact.

In FIGS. 5 and 6, a possible application for embodiments according tothe present invention in the form of both, an analysis filterbank and asynthesis filterbank will be described. One important field ofapplication is a SBR system or SBR tool (SBR=Spectral Band Replication).However, further applications of embodiments according to the presentinvention may come from other fields, in which a need for spectralmodifications (e.g. gain modifications or equalizations) exists, such asspatial audio object coding, low-delay parametric stereo coding,low-delayed spatial/surround coding, frame loss concealment, echocancellation or other corresponding applications.

The basic idea behind SBR is the observation that usually a strongcorrelation between the characteristics of a high frequency range of asignal, which will be referred to as the so-called highband signal, andthe characteristics of the lowband frequency range, further referred toas the lowband or lowband signals, of the same signal is present. Thus,a good approximation for the representation of the original input signalhighband can be achieved by a transposition from the lowband to thehighband.

In addition to the transposition, the reconstruction of the highbandincorporates shaping of spectral envelope, which comprises an adjustmentof the gains. This process is typically controlled by a transmission ofthe highband spectral envelope of the original input signal. Furtherguidance information sent from the encoder control further synthesismodules, such as an inverse filtering, a noise and sine addition inorder to cope with audio material when transposition alone might not besufficient. Corresponding parameters comprise the parameters “noisehighband” for the addition of noise and the parameter “tonalitieshighband” for the sine addition. These guidance information is usuallyreferred to as SBR data.

The SBR process can be combined with any conventional waveform or codecby means of a pre-process at the encoder side and the post-process atthe decoder side. The SBR encodes the high frequency portion of an audiosignal at a very low cost whereas the audio codec is used to code thelower frequency portion of the signal.

At the encoder side, the original input signal is analyzed, the highbandspectral envelope and its characteristics in relation to the lowband areencoded and the resulting SBR data is multiplexed with a bitstream fromthe codec for the lowband. At the decoder side, the SBR data is firstdemultiplexed. The decoding process is organized generally into steps.First, the core decoder generates the lowband and, second, the SBRdecoder operates as a post processor using the decoded SBR data to guidethe spectral band replication process. A full bandwidth output signal isthen obtained.

To obtain a coding efficiency as high as possible, and to keep thecomputational complexity low, SBR enhanced codecs are often implementedas so-called dual rate systems. Dual rate means that the band limitedcore codec is operating at half the external audio sampling rate. Incontrast, the SBR part is processed at the full sampling frequency.

FIG. 5 shows a schematic block diagram of a SBR system 500. The SBRsystem 500 comprises for instance an AAC-LD encoder (AAC-LD=AdvancedAudio Codec Low-delay) 510 and a SBR encoder 520 to which the audio datato be processed are provided in parallel. The SBR encoder 520 comprisesan analysis filterbank 530, which is shown in FIG. 5 as QMF analysisfilterbank. The analysis filterbank 530 is capable of providing subbandaudio values corresponding to subbands based on the audio signalsprovided to the SBR system 500. These subband audio values are thenprovided to a SBR parameter extraction module 540, which generates theSBR data as previously described, for instance comprising the spectralenvelope for the highband, the highband noise parameter and the highbandtonality parameter. These SBR data are then provided to the AAC-LDencoder 510.

The AAC-LD encoder 510 is in FIG. 5 shown as a dual rate encoder. Inother words, the encoder 510 operates at half the sampling frequencycompared to the sampling frequency of the audio data provided to theencoder 510. To facilitate this, the AAC-LD encoder 510 comprises adownsampling stage 550, which optionally may comprise a low pass filterto avoid distortions caused by, for instance, a violation of theNyquist-Shannon Theory. The downsampled audio data as output by thedownsampling stage 550 are then provided to an encoder 560 (analysisfilterbank) in the form of a MDCT filterbank. The signals provided bythe encoder 560 are then quantized and coded in the quantization andcoding stage 570. Moreover, the SBR data as provided by the SBRparameter extraction module 540 is also encoded to obtain a bitstream,which will then be output by the ACC-LD encoder 510. The quantizationand coding stage 570 can, for instance, quantize the data according tothe listing properties of the human ear.

The bitstream is then provided to an AAC-LD decoder 580, which is partof the decoder side to which the bitstream is transported. The AAC-LDdecoder comprises a decoding and dequantization stage 590, whichextracts the SBR data from the bitstream and the dequantized orrequantized audio data in the frequency-domain representing the lowband.The lowband data are then provided to a synthesis filterbank 600(inverse MDCT filterbank). The inverse MDCT stage (MDCT⁻¹) 600 convertsthe signals provided to the inverse MDCT stage from the frequency-domaininto the time-domain to provide a time signal. This time-domain signalis then provided to SBR decoder 610, which comprises an analysisfilterbank 620, which is shown in FIG. 5 as a QMF analysis filterbank.

The analysis filterbank 620 performs a spectral analysis of the timesignal provided to the analysis filterbank 620 representing the lowband.These data are then provided to a high frequency generator 630, which isalso referred to as a HF generator. Based on the SBR data provided bythe AAC-LD coder 580 and its decoding and dequantization stage 590, theHF generator 630 generates the highband based on the lowband signalsprovided by the analysis filterbank 620. Both, the lowband and thehighband signals are then provided to a synthesis filterbank 640, whichtransfers the lowband and highband signals from the frequency-domaininto the time-domain to provide a time-domain audio output signal formthe SBR system 500.

For the sake of completeness, it should be noted that in many cases theSBR system 500 as shown in FIG. 5 is not implemented in this way. To bemore precise, the AAC-LD encoder 510 and the SBR encoder 520 are usuallyimplemented on the encoder side, which is usually implemented separatelyfrom the decoder side comprising the AAC-LD decoder 580 and the SBRdecoder 610. In other words, the system 500 shown in FIG. 5 essentiallyrepresents the connection of two systems, namely an encoder comprisingthe aforementioned encoders 510, 520 and a decoder comprising theaforementioned decoders 580, 610.

Embodiments according to the present invention in the form of analysisfilterbanks 100 and synthesis filterbanks 300 may for instance beimplemented in the system 500 shown in FIG. 5, as a replacement of theanalysis filterbank 530, the analysis filterbank 620 and the synthesisfilterbank 640. In other words, synthesis or analysis filterbanks of theSBR components of the system 500 may, for instance, be replace bycorresponding embodiments according to the present invention. Moreover,the MDCT 560 and the inverse MDCT 600 may also be replaced by low-delayanalysis and synthesis filterbanks, respectively. In this case, if allthe described replacements have been implemented, the so-called enhancedlow-delay AAC codec (codec=coder-decoder) will be realized.

The enhanced low-delay AAC (AAC-ELD) aims at combining the low-delayfeatures of an AAC-LD (Advanced Audio Codec-Low-delay) with a highcoding efficiency of HE-AAC (High Efficiency Advanced Audio Codec) byutilizing SBR with AAC-LD. The SBR decoder 610 acts in this scenario asa post-processor, which is supplied after the core decoder 580 includinga complete analysis filterbank and a synthesis filterbank 640.Therefore, the components of the SBR decoder 610 add further decodingdelay, which is illustrated in FIG. 5 by the shading of the components620, 630, 540.

In many implementations of SBR systems 500, the lower frequency part orlowband ranges typically from 0 kHz to typically 5-15 kHz and is codedusing a waveform coder, referred to as core codec. The core codec mayfor instance be one of the MPEG audio codec family. Additionally, areconstruction of the high frequency part or highband is accomplished bya transition of the lowband. The combination of SBR with a core coder isin many cases implemented as a dual rate system, where the underlyingAAC encoder/decoder is operated at half the sampling rate of the SBRencoder/decoder.

The majority of the control data is used for the spectral enveloperepresentation, which has a varying time and frequency resolution to beable to control the SBR process as best as possible with as littlebitrate overhead as possible. The other control data mainly strives tocontrol the tonal-to-noise ratio of the highband.

As shown in FIG. 5, the output from the underlying AAC decoder 580 istypically analyzed with a 32-channel QMF filterbank 620. Then, theHF-generator module 630 recreates the highband by patching QMF subbandsfrom the existing lowband to the highband. Furthermore, inversefiltering is done on a per subband basis, based on the control dataobtained from the bitstream (SBR data). The envelope adjuster modifiesthe spectral envelope of the regenerated highband and adds additionalcomponents such as noise and sinusoids are added according to thecontrol data in the bitstream. Since all operations are done in thefrequency-domain (also known as QMF or subband domain), the final stepof the decoder 610 is a QMF synthesis 640 to retain a time-domainsignal. For instance, in the case that the QMF analysis on the encoderside is done on a 32 QFM subband system for 1024 time-domain samples,the high frequency reconstruction results in 64-QMF subbands upon whichthe synthesis is done producing 2048 time-domain samples, so that anupsampling by a factor of 2 is obtained.

In addition, the delay of the core coder 510 is doubled by operating athalf of the original sampling rate in the dual rate mode, which givesrise to additional sources of delay in both, the encoder and the decoderprocess of a AAC-LD in combination with SBR. In the following, suchdelay sources are examined and their associated delay is minimized.

FIG. 6 shows a simplified block diagram of the system 500 shown in FIG.5. FIG. 6 concentrates on delay sources in the encoder/decoder processusing SBR and low-delay filterbanks for coding. Comparing FIG. 6 withFIG. 5, the MDCT 560 and the inverse MDCT 600 have been replaced bydelay optimized modules, the so-called low-delay MDCT 560′ (LD MDCT) andthe low-delay inverse MDCT 600′ (LD IMDCT). Moreover, the HF-generator630 has also been replaced by a delay optimized module 630′.

Apart from the low-delay MDCT 560′ and the low-delay inverse MDCT 600′,a modified SBR framing and a modified HF generator 630′ are employed inthe system shown in FIG. 6. In order to avoid delay by different framingof a core coder/decoder 560, 600 and the respective SBR modules, the SBRframing is adapted to fit the framing length of 480 or 512 samples ofthe AAC-LD. Furthermore, the variable time grid of the HF generator 630,which implies 384 samples of delay, is restricted regarding thespreading of SBR data over adjacent AC-LD frames. Thus, the onlyremaining sources of delay in the SBR module are the filterbanks 530,620 and 640.

According to the situation depicted in FIG. 6, representing a partialimplementation of the AAC-ELD codec, some delay optimizations havealready been implemented including the use of a low-delay filterbank inthe AAC-LD core and the removal of a previously mentioned SBR overlap.For further delay improvements, the remaining modules need to beinvestigated. FIG. 6 shows the delay sources in the encoder/decoderprocess using SBR and the low-delay filterbanks called LD-MDCT andLD-IMDCT here. Compared to FIG. 5, in FIG. 6 every box represents adelay source, wherein the delay optimize modules are drawn in a shadedmanner. The like modules have not been optimized for low-delay so far.

FIG. 7 a illustrates a flowchart comprising a C- or C++-pseudo code toillustrate an embodiment according to the present invention in the formof an analysis filterbank or a corresponding method for generating audiosubband values in audio subband channels. To be even more precise, FIG.7 a represents a flowchart of a complex-valued analysis filterbank for32 bands.

As outlined before, the analysis filterbank is used to split thetime-domain signal, for instance output from the core coder into N=32subband signals. The output from the filterbank, the subband samples oraudio subband values, are in the case of a complex-valued analysisfilterbank complex-valued and thus oversampled by a factor of 2,compared to a real-value filterbank. The filtering involves andcomprises the following steps, wherein an array x(n) comprises exactly320 time-domain samples. The higher the index of the samples n into thearray is, the older the samples are.

After a start of the embodiments of the method in step S100, first, thesamples in the array x(n) are shifted by 32 positions in step S110. Theoldest 32 samples are discarded and 32 new samples are stored inpositions 31 to 0 in step S120. As shown in FIG. 7 a, the incomingtime-domain audio samples are stored in positions corresponding to adecreasing index n in the range of 31 to 0. This results in atime-reversal of the samples stored in the corresponding frame or vectorso that reversing the index of the window function to obtain theanalysis window function based on the (equally long) synthesis windowfunction has already been taken care of.

During a step S130, window coefficients ci(j) are obtained by a linearinterpolation of the coefficients c(j) based on equation (15). Theinterpolation is based on a block size (block length or number ofsubband values) of N=64 values and based on a frame comprising T=10blocks. Hence, the index of the window coefficients of the interpolatedwindow function are in the range between 0 and 319 according to equation(15). The window coefficients c(n) are given in the table in Annex 1 ofthe description. However, depending on implementational details, toobtain the window coefficients based on the values given in the tablesin the Annexes 1 and 3, additional sign changes with respect to thewindow coefficients corresponding to the indices 128 to 255 and 384 to511 (multiplication with factor (−1)) should be considered.

In these cases, the window coefficients w(n) or c(n) to be used may beobtained according tow(n)=w _(table)(n)·s(n)  (16a)with the sign switching function s(n) according to

$\begin{matrix}{{s(n)} = \begin{Bmatrix}{- 1} & {{{for}\mspace{14mu} 128} \leq n \leq {255\mspace{14mu}{and}\mspace{14mu} 384} \leq n \leq 511} \\{+ 1} & {else}\end{Bmatrix}} & \left( {16b} \right)\end{matrix}$for n=0 to 639, wherein w_(table)(n) are the values given in the tablesin the Annexes.

However, the window coefficients do not need to be implemented accordingto the table in Annex 1 to obtain, for instance, the already describedreduction of delay. To achieve this reduction of delay, whilemaintaining the quality level of the processed audio data, or to achieveanother trade-off, the window coefficients c(n) for the windowcoefficient index n in the range between 0 and 639, may fulfill one ofthe sets of relations as given in one of the Annexes 2 to 4. Moreover,it should be noted that also other window coefficients c(n) may beemployed in embodiments according to the present invention. Naturally,also other window functions comprising a different number of windowcoefficients than 320 or 640 can be implemented, although the tables inthe Annexes 1 to 4 only apply to window functions having 640 windowcoefficients.

The linear interpolation according to S130 leads to a significantquality improvement and aliasing effects reduction or cancellation inthe case of a window function comprising an even number of windowcoefficients. It should further be noted that the complex unit is not jas in the equations (1), (2) and (16), but is denoted by i=√{square rootover (−1)}.

In step S140, the samples of the array x(n) are then multipliedelement-wise by the coefficients ci(n) of the interpolated window.

In step S150, the windowed samples are summed up according to theequation given in the flowchart in FIG. 7 a to create the 64-elementarray u(n). In step S160, 32 new subband samples or audio subband valuesW(k,l) are calculated according to the matrix operation Mu, wherein theelement of the matrix M are given by

$\begin{matrix}{{{M\left( {k,n} \right)} = {2 \cdot {\exp\left( \frac{{\mathbb{i}} \cdot \pi \cdot \left( {k + 0.5} \right) \cdot \left( {{2 \cdot n} - 95} \right)}{64} \right)}}},\left\{ \begin{matrix}{0 \leq k < 32} \\{{0 \leq n < 64},}\end{matrix} \right.} & (17)\end{matrix}$wherein exp( ) denotes the complex exponential function and, aspreviously mentioned, i is the imaginary unit. Before the loop of aflowchart ends with step S170, each of the subband values W(k,l)(=W[k][l]) may be output, which corresponds to the subband sample l inthe subband having the index k. In other words, every loop in theflowchart shown in FIG. 7 a produces 32 complex-valued subband values,each representing the output from one filterbank subband.

FIG. 7 b illustrates the step S150 of collapsing the frame 150 ofwindowed time-domain audio samples comprising 10 blocks 160-1, . . . ,160-10 of windowed time-domain audio samples z(n) to the vector u(n) bya 5-fold summing of two blocks of the frame 150 each. The collapsing orretracting is done on an element-wise basis so that the windowedtime-domain audio samples corresponding to the same sample index insideeach of the blocks 160-1, 160-3, 160-5, 160-7 and 160-9 are added toobtain the corresponding value in the first blocks 650-1 of the vectoru(n). Accordingly, based on the blocks 160-2, 160-4, 160-6, 160-8 and160-10 the corresponding elements of the vector u(n) in block 160-2 aregenerated in step S150.

A further embodiment according to the present invention in the form ofan analysis filterbank can be implemented as a 64-band complex low-delayfilterbank. The processing of this complex low-delay filterbank as ananalysis filterbank is basically similar to the analysis filterbank asdescribed in the context of FIG. 7 a. Due to the similarities andbasically the same processing as described in the context of FIG. 7 a,the differences between the described complex analysis filterbank for 32bands of FIG. 7 a and the complex analysis filterbank for 64 subbandswill be outlined here.

In contrast to the 32-subband comprising analysis filterbank as shown inFIG. 7 a, the vector of frame x(n) comprises, in the case of a 64-bandanalysis filterbank 640 elements having indices from 0-639. Hence, thestep S110 is modified such that the samples in the array x(n) areshifted by 64 positions, wherein the oldest 64 samples are discarded. Instep S120 instead of 32 new samples, 64 new samples are stored in thepositions 63 to 0. As shown in FIG. 7 c, the incoming time-domain audiosamples are stored in positions corresponding to a decreasing index n inthe range of 63 to 0. This results in a time-reversal of the samplesstored in the corresponding frame or vector so that reversing the indexof the window function to obtain the analysis window function based onthe (equally long) synthesis window function has already been taken careof.

As the window c(n) used for windowing the elements of the vector offrame x(n), comprises typically 640 elements, the step S130 of linearlyinterpolating the window coefficients to obtain the interpolated windowsci(n) can be omitted.

Then, during step S140, the samples of the array x(n) are multiplied orwindowed by use of the sequence of window coefficients c(n), which areonce again based on the values in the table in Annex 1. In the case ofthe window coefficient c(n) are those of the synthesis window function,the windowing or multiplication of the array x(n) by the window c(n) iscarried out according to the equationz(n)=x(n)·c(n)  (18)for an=0, . . . , 639. Once again, to achieve the low-delay propertiesof the window function, it is not necessitated to implement the windowfunction exactly according to the window coefficients based on thevalues given in the table of Annex 1. For many applications, animplementation in which the window coefficients fulfill either set ofrelations as given in the tables in the Annexes 2 to 4 will besufficient to achieve an acceptable trade-off between quality and asignificant reduction of the delay. However, depending onimplementational details, to obtain the window coefficients based on thevalues given in the tables in the Annexes 1 and 3, additional signchanges with respect to the window coefficients corresponding to theindices 128 to 255 and 384 to 511 (multiplication with factor (−1))should be considered according to equations (16a) and (16b).

Step S150 of the flowchart shown in FIG. 7 a is then replaced by asumming of the samples of the vector of frame z(n) according to theequation

$\begin{matrix}{{u(n)} = {\sum\limits_{j = 0}^{5}\left( {n + {j \cdot 128}} \right)}} & (19)\end{matrix}$to create the 128-element array u(n).

Step S160 of FIG. 7 a is then replaced by a step in which 64 new subbandsamples are calculated according to the matrix operation Mu, wherein thematrix elements of the matrix M are given by

$\begin{matrix}{{{M\left( {k,n} \right)} = {2 \cdot {\exp\left( \frac{{\mathbb{i}} \cdot \pi \cdot \left( {k + 0.5} \right) \cdot \left( {{2 \cdot n} - 191} \right)}{128} \right)}}},\left\{ \begin{matrix}{0 \leq k < 64} \\{{0 \leq n < 128},}\end{matrix} \right.} & (20)\end{matrix}$wherein exp( ) denotes the complex exponential function and i is asexplained, the imaginary unit.

FIG. 7 c illustrates a flowchart according to an embodiment of thepresent invention in the form of real-valued analysis filterbank for 32subband channels. The embodiment as illustrated in FIG. 7 c does notdiffer significantly from the embodiment shown in FIG. 7 a. The maindifference between the two embodiments is that step S160 of calculatingthe new 32 complex-valued subband audio values is replaced in theembodiment shown in FIG. 7 c by a step S162 in which 32 real-valuedsubband audio samples are calculated according to a matrix operationM_(r)u, wherein the elements of the matrix M_(r) are given by

$\begin{matrix}{{{M_{r}\left( {k,n} \right)} = {2 \cdot {\cos\left( \frac{\pi \cdot \left( {k + 0.5} \right) \cdot \left( {{2 \cdot n} - 95} \right)}{64} \right)}}},\left\{ \begin{matrix}{0 \leq k < 32} \\{0 \leq n < 64.}\end{matrix} \right.} & (21)\end{matrix}$

As a consequence, every loop in the flowchart produces 32 real-valuedsubband samples wherein W(k,l) corresponds to the subband audio sample 1of the subband k.

The real-valued analysis filterbank can for instance be employed in theframework of a low-power mode of a SBR system, as shown in FIG. 5. Thelow-power mode of the SBR tool differs from the high quality SBR toolmainly with respect to the fact that real-valued filterbanks areemployed. This reduces the computational complexity and thecomputational effort by a factor of 2, so that the number of operationsper time unit are essentially reduced by a factor of 2 as no imaginarypart is necessitated to be calculated.

The proposed new filterbanks according to the present invention arefully compatible with the low-power mode of SBR systems. Thus, withfilterbanks according to the present invention, SBR systems can stillrun both in the normal mode or high-quality mode with complexfilterbanks and in the low-power mode with real-valued filterbanks. Thereal-valued filterbank may, for instance, be derived from the complexfilterbank by using only the real-values (cosine-modulatedcontributions) and omitting the imaginary values (sine-modulatedcontributions).

FIG. 8 a shows a flowchart according to a comparative example of thepresent invention in the form of a complex-valued synthesis filterbankfor 64 subband channels. As previously outlined, the synthesis filteringof the SBR-processed subband signals is achieved using a 64-subbandsynthesis filterbank. The output from the filterbank is a block ofreal-valued time-domain samples as outlined in the context of FIG. 1.The process is illustrated by the flowchart in FIG. 8 a, which alsoillustrates a comparative example in the form of a method for generatingtime-domain audio samples.

The synthesis filtering comprises after a start (step S200), thefollowing steps, wherein an array v comprises 1280 samples. In stepS210, the samples in the array v are shifted by 128 positions, whereinthe oldest 128 samples are discarded. In step S220, the 64 newcomplex-valued audio subband values are multiplied by a matrix N,wherein the matrix elements N(k,n) are given by

$\begin{matrix}{{{N\left( {k,n} \right)} = {\frac{1}{64} \cdot {\exp\left( \frac{{\mathbb{i}} \cdot \pi \cdot \left( {k + 0.5} \right) \cdot \left( {{2 \cdot n} - 63} \right)}{128} \right)}}},\left\{ \begin{matrix}{0 \leq k < 64} \\{{0 \leq n < 128},}\end{matrix} \right.} & (22)\end{matrix}$wherein exp( ) denotes the complex exponential function and is theimaginary unit. The real part of the output from this operation isstored in the position 0-127 of array v, as illustrated in FIG. 8 a.

In step S230, the samples, which are now in the time-domain areextracted from the array v according to the equation given in FIG. 8 ato create a 640-element array g(n). In step S240, the real-valuedsamples in the time-domain of array g are multiplied by the windowcoefficient c(n) to produce an array w, wherein the window coefficientsare once again the window coefficients based on the values given in thetable in Annex 1.

However, as outlined before, the window coefficients do not need to beexactly based on the values given in table of Annex 1. It is indifferent comparative examples sufficient, if the window coefficientssatisfy one of the sets of relations as given in the tables of Annexes 2to 4, to achieve the desired low-delay property of the synthesisfilterbank. Moreover, as explained in the context of the analysisfilterbank, also other window coefficients may be utilized in theframework of the synthesis filterbank. However, depending onimplementational details, to obtain the window coefficients based on thevalues given in the tables in the Annexes 1 and 3, additional signchanges with respect to the window coefficients corresponding to theindices 128 to 255 and 384 to 511 (multiplication with factor (−1))should be considered.

In step S250, 64 new output samples are calculated by a summation ofsamples from the array w(n) according to the last step and the formulagiven in the flowchart of FIG. 8 a, before one loop of a flowchart endsin step S260. In the flowchart as shown in FIG. 8 a, X[k] [l] (=X(k,l))corresponds to audio subband value l in the subband having the index k.Every new loop as depicted in FIG. 8 a produces 64 time-domain,real-valued audio samples as an output.

The implementation as shown in FIG. 8 a of a complex-valued analysisfilterbank for 64 bands does not necessitate an overlap/add buffercomprising several storage positions as explained in the context of theembodiment shown in FIG. 2 b. Here, the overlap-add buffer is “hidden”in the vectors v and g, which is calculated based on the values storedin the vector v. The overlap-add buffer is implemented in the frameworkof these vectors with these indices being larger than 128, so that thevalues correspond to values from previous or past blocks.

FIG. 8 b illustrates a flowchart of a real-valued synthesis filterbankfor 64 real-valued audio subband channels. The real-valued synthesisfilterbank according to FIG. 8 b can also implemented in the case of alow-power SBR implementation as a corresponding SBR filterbank.

The flowchart of FIG. 8 b differs form the flowchart of FIG. 8 a, mostlywith respect to step S222, which replaces S220 of FIG. 8 a. In stepS222, the 64 new real-valued audio subband values are multiplied by amatrix N_(r), wherein the elements of the matrix N_(r)(k,n) are given by

$\begin{matrix}{{{N_{r}\left( {k,n} \right)} = {\frac{1}{32} \cdot {\cos\left( \frac{\pi \cdot \left( {k + 0.5} \right) \cdot \left( {{2 \cdot n} - 63} \right)}{128} \right)}}},\left\{ \begin{matrix}{0 \leq k < 64} \\{{0 \leq n < 128},}\end{matrix} \right.} & (23)\end{matrix}$wherein the output from this operation is once again stored in thepositions 0-127 of the array v.

Apart from these modifications, the flowchart as shown in FIG. 8 b inthe case of a real-valued synthesis filterbank for the low-power SBRmode, does not differ from the flowchart as shown in FIG. 8 a of thecomplex-valued synthesis filterbank for the high quality SBR mode.

FIG. 8 c illustrates a flowchart according to an embodiment of thepresent invention in the form of a downsampled complex-valued synthesisfilterbank and the appropriate method, which can for instance beimplemented in a high-quality SBR implementation. To be more precise,the synthesis filterbank as described in FIG. 8 c relates to acomplex-valued synthesis filterbank capable of processing complex-valuedaudio subband values for 32 subband channels.

The downsampled synthesis filtering of the SBR-process subband signalsis achieved using a 32-channel synthesis filterbank as illustrated inFIG. 8 c. The output from the filterbank is a block of real-valuedtime-domain samples. The process is given in the flowchart of FIG. 8 c.The synthesis filtering comprises after a start (step S300), thefollowing steps, wherein an array v comprises 640 real-valuedtime-domain samples.

In step S310, the samples in the array v are shifted by 64 positions,wherein the oldest 64 samples are discarded. Then, in step S320, the 32new complex-valued subband samples or complex-valued audio subbandvalues are multiplied by a matrix N, the elements of which are given by

$\begin{matrix}{{{N\left( {k,n} \right)} = {\frac{1}{64} \cdot {\exp\left( \frac{{\mathbb{i}} \cdot \pi \cdot \left( {k + 0.5} \right) \cdot \left( {{2 \cdot n} - 31} \right)}{64} \right)}}},\left\{ \begin{matrix}{0 \leq k < 32} \\{{0 \leq n < 64},}\end{matrix} \right.} & (24)\end{matrix}$wherein exp( ) denotes the complex exponential function and i is againthe imaginary unit. The real part of the output from this operation isthen stored in the positions 0-63 of array v.

In step S330, the samples are extracted from vector v according to theequation given in the flowchart of FIG. 8 c to create a 320-elementarray g. In step S340, the window coefficients ci(n) of an interpolatedwindow function are obtained by a linear interpolation of thecoefficients c(n) in accordance with equation (15), wherein the index nis once again in the range between 0 and 319 (N=64, T=10 for equation(15)). As illustrated before, the coefficients of a window function c(n)are based on the values given in the table of Annex 1. Moreover, toachieve the low-delay property as illustrated earlier, the windowcoefficients c(n) do not need to be exactly the figures given in thetable of Annex 1. It is sufficient if the window coefficients c(n)fulfill at least one set of relations as given in the Annexes 2 to 4.However, depending on implementational details, to obtain the windowcoefficients based on the values given in the tables in the Annexes 1and 3, additional sign changes with respect to the window coefficientscorresponding to the indices 128 to 255 and 384 to 511 (multiplicationwith factor (−1)) should be considered according to equations (16a) and(16b). Moreover, also different window functions comprising differentwindow coefficients c(n) can naturally be employed in embodiments of thepresent invention.

In step S350, the samples of the array g are multiplied by theinterpolated window coefficient ci(n) of the interpolated windowfunction to obtain the windowed time-domain sample w(n).

Then, in step S360, 32 new output samples are calculated by a summationof samples from array w(n) according to the last step S360, before thefinal step S370 in the flowchart of FIG. 8 c.

As indicated earlier, in the flowchart of FIG. 8 c, X([k] [l]) (=x(k,l))corresponds to an audio subband value l in the audio subband channel k.Moreover, every new loop of a flowchart as indicated in FIG. 8 cproduces 32 real-valued time-domain samples as an output.

FIG. 8 d shows a flowchart of an embodiment according to the presetinvention in the form of a downsampled real-valued synthesis filterbank,which can for instance be employed in the case of a low-power SBRfilterbank. The embodiment and the flowchart shown in FIG. 8 d differsfrom the flowchart shown in FIG. 8 c of the downsampled complex-valuedsynthesis filterbank only with respect to step S320, which is replacedin the flowchart shown in FIG. 8 d by step S322.

In step S322, the 32 new real-valued audio subband values, or subbandsamples are multiplied by the matrix N_(r), wherein the elements of thematrix N_(r) are given by

$\begin{matrix}{{{N_{r}\left( {k,n} \right)} = {\frac{1}{32} \cdot {\cos\left( \frac{\pi \cdot \left( {k + 0.5} \right) \cdot \left( {{2 \cdot n} - 31} \right)}{64} \right)}}},\left\{ \begin{matrix}{0 \leq k < 32} \\{{0 \leq n < 64},}\end{matrix} \right.} & (25)\end{matrix}$wherein the output from this operation is stored in the position of 0 to64 of array v.

FIG. 9 a shows an implementation of a comparative example in the form ofa method corresponding to a complex-valued analysis filterbank for 64subbands. FIG. 9 a shows an implementation as a MATLAB-implementation,which provides as an output a vector y and a vector “state”. Thefunction as defined in this script shown in FIG. 9 a is called LDFB80 towhich a vector x comprising fresh audio samples and the vector “state”is provided to as an input. The name of the function LDFB80 is anabbreviation for low-delay filterbank for 8 blocks extending into thepast and 0 blocks into the future.

In the MATLAB-programming language, the percent sign (%) indicatesremarks, which are not carried out, but merely serve the purpose ofcommenting and illustrating the source code. In the followingdescription, different segments of the source code will be explainedwith respect to their functions.

In the code sequence S400, the buffer which is represented by the vector“state” is updated in a way such that the content of the vector “state”having the indices 577 to 640 are replaced by the contents of the vectorx comprising the fresh time-domain audio input samples. In the codesequence S410, the window coefficients of the analysis window functionas stored in the variable LDFB80_win is transferred to the vectorwin_ana.

In step S420, which assumes that the latest samples are aligned to theright side of the buffer, the actual windowing is performed. In blockS420, the content of the vector state is element-wise multiplied (·*)with the elements of the vector win_ana comprising the analysis windowfunction. The output of this multiplication is then stored into thevector x_win_orig.

In step S430, the content of the vector x_win_orig is reshaped to form amatrix of a size of 128·5 elements called x_stack. In step S440, thesign change of the stack x_stack is performed with respect to the secondand fourth column of the matrix x_stack.

In step S450, the stack x_stack is collapsed or retracted by summing theelements of x_stack with respect to the second index and simultaneouslyinverting the order of the elements and transposing the outcome beforestoring the outcome again to the various x_stack.

In the code segment S460, the transformation from the time-domain intothe frequency-domain is carried out by computing a complex Fast Fouriertransformation (FFT) of the element-wise multiplied content of the stackx_stack multiplied with the complex exponential function to which theargument (−i·π·n/128) is provided, with the indices and in the rangefrom 0 to −127 and the imaginary unit i.

In the code segment S470, a post twiddle is performed by defining thevariable m=(64+1)/2 and by calculating the block comprising the audiosubband values as a vector y according to the equation

$\begin{matrix}{{y(k)} = {\overset{\_}{2 \cdot {{temp}(k)} \cdot {\exp\left( {{- 2}\;{{\mathbb{i}} \cdot \pi \cdot \left( {\left( {k - 1 + \frac{1}{2}} \right) \cdot \frac{m}{128}} \right)}} \right)}}.}} & (26)\end{matrix}$

The index k covers the range of integers from 1-64 in the implementationshown in FIG. 9 a. The vector y is then output as the vector or blockcomprising the audio subband values 180 of FIG. 1. The bar above thesecond factoring equation (26) as well as the function conj( ) encodesegment S417 in FIG. 9 a refer to the complex conjugate of the argumentof the respective complex number.

In a final code-segment S480, the state-vector is shifted by 64elements. The state-vector in its shifted form may then be provided tothe function LDFB80 as an input again in a further loop of the function.

FIG. 9 b shows a MATLAB-implementation according to an embodiment of thepresent invention in the form of a method corresponding to acomplex-valued analysis filterbank for 32 subbands. Accordingly, thefunction defined is referred to as LDFB80_(—)32 indicating that theimplementation represents a low-delay filterbank for 32 subbands basedon an additional overlap of 8 blocks into the past and 0 blocks into thefuture.

The implementation of FIG. 9 b differs from the implementation shown inFIG. 9 a, only with respect to a few code sequences, as will be outlinedin the following description. The code sequences S400, S430, S460, S470and S480 are replaced by corresponding code sequences S400′, S430′,S460′, S470′ and S480′ taking into account mainly the fact that thenumber of subbands, or the number of subband values output by thefunction LDFB80_(—)32, is reduced by a factor of 2. Accordingly, thestep S400′ relates to the vector state being updated with respect to the32 last entries corresponding to the indices 289 to 320 with thecorresponding 32 time-domain audio input samples of the fresh block 220as shown in FIG. 1.

However, the major difference between the implementations as shown inFIGS. 9 a and 9 b appears in the code sequence S410 of FIG. 9 a, whichis replaced by a code sequence S412 in the implementation shown in FIG.9 b. The code sequence for S412 of FIG. 9 b comprises first a copying ofthe 640 window coefficients comprising windows stored in the vectorLDFB80 win to the local vector win_ana. Then, an interpolation accordingto equation (15) takes place, in which two consecutive windowcoefficients represented by the vector elements of the vector win_anaare added and divided by 2 and then stored back in the vector win_ana.

The next code sequence S420 is identical to the code sequence S420 asshown in FIG. 9 a, which carries out the actual element-wisemultiplication (·*) of the windowing of the values, or elements, of thevector state with the elements of the vector win_ana comprising theinterpolated window coefficients of the interpolated window function.The output of this operation is stored in the vector x_win_orig.However, the difference between the code sequence S420 of FIG. 9 b andthe corresponding code sequence S420 of FIG. 9 a, is that in the case ofFIG. 9 b, not 640 but only 320 multiplications are carried out in theframework of the windowing.

In the code sequence S430′ replacing the code sequence S430, the stackx_stack is prepared by reshaping the vector x_win_orig. However, as thevector X_win_orig only comprises 320 elements, compared to thecorresponding vector of FIG. 9 a comprising 640 elements, the matrixx_stack is only a matrix of 64·5 elements.

The code sequences S440 of the sign change and the code sequence S450 ofcollapsing the stack are identical in both implementations according tothe FIGS. 9 a and 9 b, apart from the reduced number of elements (320compared to 640).

In the code sequence S460′ replacing the code sequence S460 an oddcomplex Fast Fourier Transform (FFT) of a window data is carried out,which is quite similar to the transform of code sequence S460 of FIG. 9a. However, once again, due to the reduced number of output audiosubband values, the vector temp is provided with the outcome of a FastFourier Transform, the element-wise multiplication of the elements ofthe stack x_stack and the complex exponential function of the argument(−i·π·n/64), wherein the index n is in the range between 0 and 63.

Afterwards, in the modified code sequence S470′, the post twiddling isperformed by defining the variable m=(32+1)/2 and by generating theoutput vector y according to equation (26), wherein the index k onlycovers the range from 1 to 32 and wherein the number 128 appearing inthe argument of the complex exponential function is substituted by thenumber 64.

In the final code-sequence S480′, the buffer state is shifted by 32elements in the case of the implementation shown in FIG. 9 b, wherein inthe corresponding code sequence S480, the buffer is shifted by 64elements.

FIG. 10 a shows a MATLAB-script illustrating an implementation accordingto a comparative example in the form of a method corresponding to acomplex-valued synthesis filterbank for 64 subbands. The script shown inFIG. 10 a defines the function ILDFB80 to which the vector xrepresenting the block 320 of audio subband values of FIG. 2 a and astate-vector “state” is provided to as input parameters. The nameILDFB80 indicates that the function defined is an inverse low-delayfilterbank corresponding to 8 blocks of audio data from the past and 0blocks from the future. The function provides a vector y and a new orredefined state-vector “state” as an output, wherein the vector ycorresponds to the block 410 of time-domain audio samples from FIG. 2 a.

In a code sequence S500, a pre-twiddling is performed, in which avariable m=(64+1)/2 as well as a vector temp is defined. The elementstemp(n) of the vector temp are defined according to equation

$\begin{matrix}{{{{temp}(n)} = {\frac{1}{2} \cdot \overset{\_}{x(n)} \cdot {\exp\left( {2\;{{\mathbb{i}} \cdot {\pi\left( {n - 1 + \frac{1}{2}} \right)} \cdot \frac{m}{128}}} \right)}}},} & (27)\end{matrix}$wherein the bar above the element of the vector x(n) and the functionconj( ) represent the complex conjugate, exp( ) represents the complexexponential function, i represents the imaginary unit and n is an indexin the range from 1- to 64.

In the code sequence S510, the vector temp is expended into a matrixcomprising in the first column the elements of the vector temp and inthe second column, the complex conjugate of the reversed vector tempwith respect to the order of the elements as defined by the index of thevector. Hence, in the code sequence S510 an odd symmetry of the matrixtemp is established based on the vector temp.

In a code sequence S520 an odd Fast Fourier Transform (FFT) is performedbased on the matrix temp. in this code sequence, the real part of theelement-wise multiplication of the outcome of the inverse FourierTransform of the matrix temp with the exponential function having theargument of (i·π/128) is performed and outputted to a vector y_knl,wherein the index n is in the range from 0 to 127.

In the code sequence S530, an extension of the data and an alternatingsign flip is formed. To achieve this, the order of the elements of thevector y_knl is reversed and at the same time a sign flip if carriedout. Then, a matrix tmp is defined, comprising the first, third andfifth column of the vector y_knl, wherein the second and the fourthcolumn comprise the sign flipped vector y_knl.

In a code sequence S540, the window coefficients as stored in the vectorLDFB80_win are first copied to the vector win_ana. Then, the synthesiswindow coefficients are determined based on the analysis windowcoefficients as stored in the vector win_ana by generating a timereversed version of the analysis window function according towin_syn(n)=win_ana(N·T−n)  (28)wherein N·T is the total number of window coefficients and n is theindex of the window coefficients.

In a code sequence S550, the synthesis window is applied to the vectortmp by an element-wise multiplication of the vector with the synthesiswindow function. In a code sequence S560, the buffer is updated bysetting the elements of the vector state with the indices 577 to 640 to0 and by adding the content of the windowed vector tmp to thestate-vector state.

In a code sequence S570, the output vector y comprising the time-domainaudio samples is extracted from the state-vector by extracting theelements of the state-vector by extracting the elements of thestate-vector with the indices 1 to 64.

In a code sequence S580, the final code sequence of the function asshown in FIG. 10 a, the state-vector state is shifted by 64 elements sothat the elements with indices from 65 to 640 are copied to the first576 elements of the vector state.

FIG. 10 b shows a MATLAB-script of an implementation according to anembodiment of the present invention in the form of a complex-valuedsynthesis filterbank for 32 subband values. The name of the function asdefined by the script shown in FIG. 10 b illustrates this as thefunction defined is called ILDFB80_(—)32 indicating that the functiondefined is an inverse low-delay filterbank for 32 bands with 8 blocksoverlap from the past and 0 blocks overlap from the future.

As discussed with respect to the comparison of the implementation shownin FIGS. 9 a and 9 b, the implementation according to the script of FIG.10 b is also closely related to the implementation of the 64-subbandsynthesis filterbank according to FIG. 10 a. As a consequence, the samevectors are provided to the function and are output by the functionwhich, however, comprise only half the number of elements compared tothe implementation of FIG. 10 a. The implementation for a 32-bandsynthesis filterbank for 32 bands differs from the 64-subband versionillustrated in FIG. 10 a, mainly with respect to two aspects. The codesequences S500, S510, S520, S53β, S560, S570 and S580 are replaced withcode sequences in which the number of elements to be addressed andfurther number of element-related parameters are divided by 2. Moreover,the code sequence S540 of generating the synthesis window function isreplaced by a code sequence S542, in which the synthesis window functionis generated as a linearly interpolated synthesis window functionaccording to equation (15).

In the code sequence S500′ replacing the code sequence S500, thevariable m is defined to be equal to m=(32+1)/2 and the vector temp isdefined according to equation (27), wherein the index n only covers therange of 1 to 32 and wherein the factor of 1/128 is replaced by thefactor 1/64 in the argument of the exponential-function.

Accordingly, in the code sequence S510′ replacing the code sequenceS510, the index range only covers the indices of the 32-elementcomprising the vector temp. In other words, the index only covers thevalues from 1 to 32. Accordingly, in the code sequence S520′ replacingthe code sequence S520, the argument of the exponential function isreplaced by (i·π·n/64), wherein the index n is in the range from 0 to63. In the framework of the code sequence S530′, the index range is alsoreduced by a factor of 2 compared to the code sequence S530.

The code sequence S542 replacing the code sequence S540 of FIG. 10 aalso copies the window function as stored in the vector LDFB80_win tothe vector win_ana and generates a time-reversed version win_synaccording to equation (28). However, the code sequence S542 of theimplementation shown in FIG. 10 b further comprises an interpolationstep according to equation (15), in which for each element of theredefined vector win_syn comprising the window coefficients of thesynthesis window function, a linear interpolation of two consecutivewindow coefficients of the original synthesis window function.

The code sequence S550 of applying the window to the vector tmp andreplacing the elements tmp with the windowed version thereof isidentical in terms of the code as a direct comparison of the respectivecode sequences in FIGS. 10 a and 10 b. However, due to the smaller sizeof the vector tmp in the implementation of FIG. 10 b, during animplementation, only half the number of multiplications is carried out.

Also in the framework of the code sequences S560′, S570′ and S580′replacing the code sequences S560, S570 and S580, respectively, theindices 640 and 64 are replaced by 320 and 32, respectively. Therefore,these three final code sequences only differ from the code sequences ofthe implementation shown in FIG. 10 a with respect to the size of thevector states tmp and y.

As the embodiments described so far have illustrated, the analysiswindower as well as the synthesis windower are adapted to windowing therespective samples in the time-domain comprised in the respective framesby multiplying these on an element-wise basis with window coefficientsof a window function.

Before describing a window function, which can be employed for instanceas a synthesis window function and as an analysis window function in itstime-reversed version more closely, advantages of embodiments accordingto the present invention will be outlined in more detail, especially inview of an implementation in the framework of a SBR tool or system asshown in FIGS. 5 and 6.

Among the advantages, embodiments according to the present invention andsystems comprising more than one embodiment according to the presentinvention may offer is a significant reduction of the delay according toother filterbanks. However, this low-delay property will be addressed inthe context of FIGS. 13 and 14 in more detail. One important aspect inthis context is to note that the length of the window function, in otherwords, the number of window coefficients to be applied to a frame or ablock of time-domain samples is independent of the delay.

Embodiments according to the present invention offer the furtheradvantage of improving the quality of the (reconstructed) audio data.The interpolation employed in embodiments according to the presentinvention offers a significantly reduced aliasing compared to otherreduction schemes concerning the number of window coefficients.

Moreover, as will be outlined in the context of FIGS. 17 and 18 in moredetail, in terms of the psychoacoustics, embodiments according to thepresent invention often make use the temporal masking properties of thehuman ear better than many other filterbanks. Moreover, as will be moreclosely outlined in the context of FIGS. 15, 16 and 19, embodimentsaccording to the present invention offer an excellent frequencyresponse.

Also, in many filterbanks according to an embodiment of the presentinvention, a perfect reconstruction is achievable if an analysisfilterbank and the synthesis filterbank are interconnected. In otherwords, embodiments according to the present invention do not only offeran audibly indistinguishable output compared to the input of such aninterconnected set of an analysis filterbank and a synthesis filterbank,but (apart from quantization errors, computational rounding effects andfurther effects caused by the discretization, an identical outputcompared to the input.

An integration in the SBR module of filterbanks according to the presentinvention can easily be achieved. While typically SBR modules operate inthe dual-rate mode, the complex-valued low-delay filterbanks accordingto embodiments of the present invention are capable of providing perfectreconstruction in the single-rate mode, while the original SBR QMFfilterbanks are capable of only providing near-perfect reconstruction.In the dual-rate mode, the 32-band version of the impulse response isobtained by linear interpolation also referred to as downsampling of twoadjacent taps or window coefficients of the 64-band impulse response orwindow function as explained in the context of FIG. 3.

In the case of a complex-valued implementation of a filterbank, asignificant reduced analyzing (or synthesizing) delay for criticallysampled filterbanks can be achieved, in which the sampling or processingfrequency corresponds to the border frequency according to theNyquist-Shannon Theory. In the case of a real-valued implementation of afilterbank, an efficient implementation can be achieved employingoptimized algorithms, as for instance illustrated in the context of theMATLAB-implementation shown in FIGS. 9 and 10. These implementations mayfor instance be employed for the low-power mode of the SBR tool asdescribed in the context of FIGS. 5 and 6.

As outlined in the context of FIGS. 5 and 6, it is possible to achieve afurther reduction concerning the delay in the case of an SBR system byusing a complex-valued low-delay filterbank according to an embodimentof the present invention. As outlined before, in the SBR decoder 610 asshown in FIG. 5, the QMF analysis filterbank 620 is replaced by acomplex low-delay filterbank (CLDFB) according to an embodiment of thepresent invention. This replacement can be done in a computable way bykeeping the number of bands (64), the length of the impulse response(640) and by using a complex modulation. The delay achieved by this toolis minimized to such an extent to achieve an overall delay low enoughfor a bi-directional communication without sacrificing an achievablequality level.

Compared, for instance, to a system comprising a MDCT and a MDST to forma complex-valued MDCT-like system, an embodiment according to thepresent invention provides a far better frequency response. Compared tothe QMF filterbank, for instance, used in the MPEG-4 SBR today, a systemcomprising one or more filterbanks according to embodiments of thepresent invention provides a significantly lower delay.

Even compared to a low-delay QMF filterbank, embodiments according tothe present invention offer the advantage of a perfect reconstructioncombined with the lower delay. The advantages arising from the perfectreconstruction property in contrast to the near-perfect reconstructionof QMF filterbanks are the following. For near-perfect reconstruction, ahigh stopband attenuation is needed to attenuate the aliasing to asufficiently low level. This restricts the possibility of achieving avery low-delay in the filter design. In contrast, employing anembodiment according to the present invention now has the possibility ofindependently designing the filter so that no high stopband attenuationis needed to attenuate the aliasing to sufficiently low levels. Thestopband attenuation has just to be low enough to allow reduced aliasingsufficient for the desired signal processing application. Thus, a bettertrade-off towards lower delay can be achieved in the filter design.

FIG. 11 shows a comparison of a window function 700 as can, forinstance, be employed in an embodiment according to the presentinvention along with the sine-window function 710. The window function700, which is also referred to as a “synthesis” CMLDFB-window(CMLDFB=complex modulated low-delay filterbank), comprises 640 windowcoefficients based on the values given in the table in Annex 1.Concerning the magnitude of the window functions, it should be notedthat general amplification factors or damping factors for adjusting anamplitude of the windowed signal are not considered in the following.Window functions can, for instance, be normalized with respect to avalue corresponding to the center of delay, as outlined in the contextof FIG. 13, or with respect to a value n=N, n=N−1 or n=N+1, wherein N isthe block length and n is the index of the window coefficients. Incomparison, the sine-window function 710 is only defined over 128samples and is, for instance, employed in the case of an MDCT or an MDSTmodule.

However, depending on implementational details, to obtain the windowcoefficients based on the values given in the tables in the Annexes 1and 3, additional sign changes with respect to the window coefficientscorresponding to the indices 128 to 255 and 384 to 511 (multiplicationwith factor (−1)) should be considered according to equations (16a) and(16b).

Before discussing the differences of the two window functions 700, 710,it should be noted that both window functions comprise real-valuedwindow coefficients only. Moreover, in both cases, an absolute value ofthe window coefficient corresponding to an index n=0 is smaller than0.1. In the case of a CMLDFB-window 700, the respective value is evensmaller than 0.02.

Considering the two window functions 700, 710 with respect to theirdefinition sets, several significant differences are evident. Whereasthe sine-window function 710 is symmetric, the window function 700 showsan asymmetric behavior. To define this more clearly, the sine-windowfunction is symmetric as a real-valued value n₀ exists so that withrespect to all real numbers n, so that the window function 710 isdefined for (n₀+n) and (n₀−n), the relation|w(n ₀ −n)|=|w(n ₀ +n)|  (29)is fulfilled to a desirable margin (ε≧0; the absolute value of thedifference of the terms on the two sides of equation (29) is smallerthan or equal to ε), wherein w(n) represents the window coefficientcorresponding to the index n. In the case of the sine-window therespective index n₀ is exactly in the middle of the two topmost windowcoefficients. In other words, for the sine-window 710 the index isn₀=63.5. The sine-window function is defined for indices n=0, . . . ,127.

In contrast, the window function 700 is defined over the set of indicesn=0, . . . , 639. The window function 700 is clearly asymmetric in thesense that for all real-valued numbers n₀ at least one real numberexists so that (n₀+n) and (n₀−n) belong to the definition set of thewindow function, for which the inequality|w(n ₀ −n)|≠|w(n ₀ +n)|  (30)holds to an (almost deliberately) definable margin (ε≧0; the absolutevalue of the difference of the terms on the two sides of equation (29)is greater than or equal to ε), wherein once again w(n) is the windowcoefficient corresponding to the index n.

Further differences between the two window functions, which both relateto block sizes of N=64 samples, is that the maximum value of the windowfunction 700 are larger than 1 and is acquired for indices in the rangeofN≦n≦2N  (31)for the synthesis window. In the case of the window function 700 shownin FIG. 11, the maximum value acquired is larger than 1.04 acquired atthe sample index n=77. In contrast, the maximum values of thesine-window 710 is smaller than or equal to 1, which is acquired at n=63and n=64.

However, also the window function 700 acquires a value of approximately1 at sample indices around n=N. To be more precise, the absolute valueor the value itself of the window coefficient w(N−1) corresponding tothe index n=N−1 is smaller than 1, whereas the absolute value or thevalue itself of the window coefficient w(N) corresponding to the indexn=N is larger than 1. In some embodiments according to the presentinvention, these two window coefficients obey the relations0.99<w(N−1)<1.01.0<w(N)<1.01,  (32)which is a result of optimizing the audio quality of the filterbanksaccording to embodiments of the present invention. In many cases it isdesirable to have a window coefficient w(0) comprising an absolute valueas small as possible. In this case, a determinant of the windowcoefficients|w(0)·w(2N−1)−w(N−1)·w(N)|≈1  (33)should be as close as possible to 1 to achieve an audio quality, whichis optimized with respect to the possible parameters. The sign of thedeterminant as given by equation (33) is, however, freely choosable. Asa consequence of the window coefficient w(0) being smaller orapproximately 0, the product of w(N−1) w(N) or its absolute valuesshould be as close as possible to +/−1. In this case, the windowcoefficient w(2N−1) can then be chosen almost freely. Equation (33) is aresult of employing the technique of zero-delay matrices as described in“New Framework for Modulated Perfect Reconstruction Filter Banks” by G.D. T. Schuller and M. J. T. Smith, IEEE Transactions on SignalProcessing, Vol. 44, No. 8, August 1996.

Furthermore, as will be outlined in more detail in the context of FIG.13, the window coefficients corresponding to the indices N−1 and N arecomprised in the middle of the modulation core and therefore correspondto the sample having a value of approximately 1.0 and which coincideswith the delay of the filterbank as defined by the prototype filterfunction or the window function.

The synthesis window function 700 as shown in FIG. 11 furthermore showsan oscillating behavior with strictly monotonic increasing windowcoefficients from the window coefficient of the sequence of windowcoefficients corresponding to the index (n=0) used for windowing thelatest time-domain audio sample up to the window coefficient comprisingthe highest absolute value of all window coefficients of the synthesiswindow function 700. Naturally, in the case of the time-reversedanalysis window function, the oscillating behavior comprises a strictlymonotonic decrease of the window coefficients from the windowcoefficient comprising the highest absolute value of all windowcoefficients of a corresponding (time-reverse) analysis window functionto the window coefficients of the sequence of window coefficientscorresponding to an index (n=639) used for windowing the latesttime-domain audio sample.

As a consequence of the oscillating behavior, the development of thesynthesis window function 700 starts with a window coefficientcorresponding to the index n=0 having an absolute value smaller than0.02 and an absolute value of the window coefficient corresponding tothe index n=1 of lower than 0.03, acquiring a value of about 1 at anindex n=N, acquiring a maximum value of more than 1.04 at an indexaccording to equation (31), acquiring a further value of approximately 1at an index n=90 and 91, a first sign change at the index values ofn=162 and n=163, acquiring a minimal value of less than −0.1 or −0.12755at an index of approximately n=3N and a further sign change at indexvalues n=284 and n=285. However, the synthesis window function 700further may comprise further sign changes at further index values n.When comparing the window coefficients to the values given in the tablesin the Annexes 1 and 3, the additional sign changes with respect to thewindow coefficients corresponding to the indices 128 to 255 and 384 to511 (multiplication with factor (−1)) should be considered according toequations (16a) and (16b).

The oscillating behavior of the synthesis window function 700 is similarto that of a strongly damped oscillation, which is illustrated by themaximum value of about 1.04 and the minimum value of about −0.12. As aconsequence, more than 50% of all window coefficients comprise absolutevalues being smaller than or equal to 0.1. As outlined in the context ofthe embodiments described in FIGS. 1 and 2 a, the development of thewindow function comprises a first group 420 (or 200) and a second group430 (or 210), wherein the first group 420 comprises a first consecutiveportion of window coefficients and the second group 430 comprises aconsecutive second portion of window coefficients. As already outlinedbefore, the sequence of window coefficients of the window comprises onlythe first group 420 of window coefficients and the second group ofwindow functions 430, wherein the first group 420 of window coefficientsexactly comprises the first consecutive sequence of window coefficients,and wherein the second group 430 exactly comprises the secondconsecutive portion of window coefficients. Hence, the terms first group420 and first portion of window coefficients as well as the terms secondgroup 430 and second portion of window coefficients can be usedsynonymously.

The more than 50% of all window coefficients having absolute valuessmaller than or equal to 0.1 are comprised in the second group or secondportion 430 of window coefficients as a consequence of the stronglydamped oscillatory behavior of the window function 700. Moreover, alsomore than 50% of all window coefficients comprised in the second groupor second portion 430 of window coefficients comprise absolute values ofless than or equal to 0.01.

The first portion 420 of window coefficients comprises less than onethird of all window coefficients of the sequence of window coefficients.Accordingly, the second portion 430 of window coefficients comprisesmore than two thirds of window coefficients. In the case of a totalnumber of blocks T to be processed in one of the frames 120, 150, 330,380 of more than four blocks, the first portion typically comprises3/2·N window coefficients, wherein N is the number of time-domainsamples of one block. Accordingly, the second portion comprises the restof the window coefficients or, to be more precise, (T−3/2)N windowcoefficients. In the case of T=10 blocks per frame as shown in FIG. 11,the first portion comprises 3/2·N window coefficients, whereas thesecond portion 210 comprises 8.5·N window coefficients. In the case of ablock size of N=64 time-domain audio samples per block, the firstportion comprises 96 window coefficients, whereas the second portioncomprises 544 window coefficients. The synthesis window function 700 asshown in FIG. 11 acquires a value of approximately 0.96 at the border ofthe first portion and the second portion at an index of around n=95 or96.

Despite the number of window coefficients comprised in the first portion420 and the second portion 430, an energy value or a total energy valueof corresponding window coefficients differ significantly from oneanother. The energy value as defined by

$\begin{matrix}{{E = {\sum\limits_{n}{{w(n)}}^{2}}},} & (34)\end{matrix}$wherein w(n) is a window coefficient and the index n over which the sumin equation (34) is evaluated corresponds to the indices of therespective portions 420, 430, the whole set of window coefficients orany other set of window coefficients to which the respective energyvalues E corresponds. Despite the significant difference of windowcoefficients, the energy value of the first portion 420 is equal to orhigher than ⅔ of the overall energy value of all window coefficients.Accordingly, the energy value of the second portion 430 is smaller thanor equal to ⅓ of the overall energy value of all window coefficients.

To illustrate this, the energy value of the first portion 420 of thewindow coefficients of the window function 700 is approx. 55.85, whilethe energy value of the window coefficients of the second portion 430 isapprox. 22.81. The overall energy value of all window coefficients ofthe window function 700 is approx. 78.03, so that the energy value ofthe first portion 420 is approx. 71.6% of the overall energy value,while the energy value of the second portion 430 is approx. 28.4% of theoverall energy value of all window coefficients.

Naturally, equation (34) can be stated in a normalized version bydividing the energy value E by a normalization factor E₀, which can inprinciple be any energy value. The normalization factor E₀ may, forinstance, be the overall energy value of all window coefficients of thesequence of window coefficients calculated according to equation (34).

Based on the absolute values of the window coefficients or based on theenergy values of the respective window coefficients, also a center pointor a “center of mass” of the sequence of window coefficients can bedetermined. The center of mass or the center point of the sequence ofwindow coefficients is a real number and typically lies in the range ofindices of the first portion 420 of the window coefficients. In the caseof the respective frames comprising more than four blocks of time-domainaudio samples (T>4), the center of mass n_(ca) based on the absolutevalues of the window coefficients or the center of mass n_(ce) based onthe energy values of the window coefficients is smaller than 3/2·N. Inother words, in the case of T=10 blocks per frame, the center of masslies well within the region of indices of the first portion 200.

The center of mass n_(ca) based on the absolute values of the windowcoefficients w(n) are defined according to

$\begin{matrix}{n_{ca} = \frac{\sum\limits_{n = 0}^{{N \cdot T} - 1}{n \cdot {{w(n)}}}}{\sum\limits_{n = 0}^{{N \cdot T} - 1}{{w(n)}}}} & (35)\end{matrix}$and the center of mass n_(ce) in view of the energy values of the windowcoefficients w(n) are defined according to

$\begin{matrix}{{n_{ce} = \frac{\sum\limits_{n = 0}^{{N \cdot T} - 1}{n \cdot {{w(n)}}^{2}}}{\sum\limits_{n = 0}^{{N \cdot T} - 1}{{w(n)}}^{2}}},} & (36)\end{matrix}$wherein N and T are positive integers indicating the number oftime-domain audio samples per block and the number of blocks per frame,respectively. Naturally, the center points according to equations (35)and (36) can also be calculated with respect to a limited set of windowcoefficients by replacing the limits of the sums above accordingly.

For the window function 700 as shown in FIG. 1, the center of massn_(ca) based on the absolute values of the window coefficients w(n) isequal to a value of n_(ca)≈87.75 and the center point or center of massn_(ce) with respect to the energy values of the window coefficients w(n)is n_(ce)≈80.04. As the first portion 200 of window coefficients of thewindow function 700 comprises 96 (=3/2·N; N=64) window coefficients,both center points lie well within the first portion 200 of the windowcoefficients, as previously outlined.

The window coefficients w(n) of the window function 700 are based on thevalues given in the table in Annex 1. However, to achieve, for instance,the low-delay property of the filterbank as outlined before, there is noneed to implement the window function as precisely as given by thewindow coefficients in the table of Annex 1. In many cases, it is morethan sufficient for the window coefficients of a window functioncomprising 640 window coefficients to fulfill any of the relations orequations given in the tables of Annexes 2 to 4. The window coefficientsor filter coefficients given in the table in Annex 1 representadvantageous values, which might be adapted according to equations (16a)and (16b) in some implementations. However, as indicated, for instance,by the further tables given in the further Annexes, the advantageousvalues can be varied from the second, third, fourth, fifth digit afterthe decimal point so that the resulting filters or window functionsstill have the advantages of embodiments according to the presentinvention. However, depending on implementational details, to obtain thewindow coefficients based on the values given in the tables in theAnnexes 1 and 3, additional sign changes with respect to the windowcoefficients corresponding to the indices 128 to 255 and 384 to 511(multiplication with factor (−1)) should be considered according toequations (16a) and (16b).

Naturally, further window functions comprising a different number ofwindow coefficients can equally be defined and be used in the frameworkof embodiments according to the present invention. In this context itshould be noted that both the number of time-domain audio samples perblock and the number of blocks per frame as well as the distribution ofthe blocks with respect to past samples and future samples can be variedover a wide range of parameters.

FIG. 12 shows a comparison of a complex modulated low-delay filterbankwindow (CMLDFB-window) 700 as shown in FIG. 11 and the original SBR QMFprototype filter 720 as employed, for instance, in the SBR toolaccording to the MPEG standards. As shown in FIG. 11, the CMLDFB window700 is once again the synthesis window according to an embodiment of thepresent invention.

While the window function 700 according to an embodiment of the presentinvention is clearly asymmetric as defined in the context of equation(30), the original SBR QMF prototype filter 720 is symmetric withrespect to the indices n=319 and 320, as the window function 700 as wellas the SBR QMF prototype filter 720 are each defined with respect to 640indices each. In other words, with respect to equation (29) the “indexvalue” n₀ representing the index of the symmetry center is given byn₀=319.5 in the case of the SBR QMF prototype filter 720.

Moreover, due to the symmetry of the SBR QMF prototype filter 720, alsothe center point n_(ca) and n_(ce) according to equations (35) and (36),respectively, are identical to the symmetry center n₀. The energy valueof the SBR QMF prototype filter 720 is 64.00 as the prototype filter isan orthogonal filter. In contrast, the clearly asymmetric windowfunction 700 comprises an energy value of 78.0327 as outlined before.

In the following sections of the description, SBR systems as outlined inthe context of FIGS. 5 and 6 will be considered, in which the SBRdecoder 610 comprises embodiments according to the present invention inthe form of an analysis filterbank as the filterbank 620 and anembodiment according to the present invention in the form of a synthesisfilterbank for the synthesis filterbank 640. As will be outlined in moredetail, the overall delay of an analysis filterbank according to thepresent invention employing the window function 700 as shown in FIGS. 11and 12 comprises an overall delay of 127 samples, whereas the originalSBR QMF prototype filter-based SBR tool results in an overall delay of640 samples.

The replacement of the QMF filterbanks in the SBR module, for instancein the SBR decoder 610, by a complex-valued low-delay filterbank (CLDFB)results in a delay reduction from 42 ms to 31.3 ms without introducingany degradation of audio quality or additional computational complexity.With the new filterbank both, the standard SBR mode (high-quality mode)and the low-power mode employing only real-valued filterbanks, aresupported, as the description of embodiments according to the presentinvention with respect to FIGS. 7 to 10 has shown.

Especially in the field of telecommunication and bi-directionalcommunication, a low-delay is of great importance. While the enhancedlow-delay AAC is already capable of achieving a delay low enough forcommunication applications of 42 ms, its algorithmic delay is stillhigher than that of the low-delay AAC corecodec, which is capable ofachieving delays of down to 20 ms and that of other telecommunicationcodecs. In the SBR decoder 610, the QMF analysis and synthesis stagesstill cause a reconstruction delay of 12 ms. A promising approach toreduce that delay is to utilize a low-delay filterbank techniqueaccording to an embodiment of the present invention and to replace thecurrent QMF filterbanks by a respective low-delay version according tothe embodiments of the present invention. In other words, a furtherdelay reduction is achieved by simply replacing the regular filterbanksused in the SBR module 610 by a complex low-delay filterbank accordingto the embodiments of the present invention.

For the usage in the SBR module 610, the new filterbanks according toembodiments of the present invention, which are also referred to asCLDFBs, are designed to be as similar to the originally used QMFfilterbanks as possible. This includes, for instance, the use of 64subbands or bands, an equal length of the impulse responses and acompatibility with dual-rate modes as employed in SBR systems.

FIG. 13 illustrates the comparison of the CLDFB window shape 700according to an embodiment of the present invention and the original SBRQMF prototype filter 720. Furthermore, it illustrates the delay ofmodulated filterbanks, which can be determined by analyzing the overlapdelay introduced by the prototype filter or window function in additionto the framing delay of the modulation core having a length of N samplesin the case of a DCT-IV-based system. The situation shown in FIG. 13refers once again to the case of a synthesis filterbank. The windowfunction 700 and the prototype filter function 720 also representimpulse responses of the synthesis prototype filters of the twofilterbanks involved.

With respect to the delay analysis for both the SBR QMF filterbank andthe proposed CLDFB according to an embodiment of the present invention,in the analysis and the synthesis only the overlap to the right side andthe left side of the modulation core, respectively, adds delay.

For both filterbanks, the modulation core is based on a DCT-IVintroducing a delay of 64 samples, which is marked in FIG. 13 as thedelay 750. In the case of the SBR QMF prototype filter 720 due to thesymmetry the modulation core delay 750 is symmetrically arranged withrespect to the center of mass or center point of the respectiveprototype filter function 720 as indicated in FIG. 13. The reason forthis behavior is that the buffer of the SBR QMF filterbank needs to befilled up to a point that the prototype filter function 720 having themost significant contribution in terms of the respective energy valuesof the prototype filter values will be considered in the processing. Dueto the shape of the prototype filter function 720, this necessitates thebuffer to be filled up at least to the center point or center of mass ofthe respective prototype filter function.

To illustrate this further, starting from an all initialized buffer ofthe corresponding SBR QMF filterbank, the buffer needs to be filled upto a point that a processing of data will result in a processing ofsignificant data, which necessitates the respective window function orprototype filter function to have a significant contribution. In thecase of the SBR QMF prototype filter function, the symmetric shape ofthe prototype filter 720 yields a delay, which is of the order of thecenter of mass or center point of the prototype filter function.

However, as the delay introduced by the modulation core of theDCT-IV-based system of N=64 for samples is present and the system alsocomprises a delay of one block, it can be observed that the synthesisprototype for the SBR QMF introduces an overlap delay of 288 samples.

As indicated earlier, in the case of the synthesis filterbanks to whichFIG. 13 relates, this additional left-side overlap 760 causes the delay,while the right-side overlap 770 relates to past samples and thereforedoes not introduce an additional delay in the case of a synthesisfilterbank.

In contrast, starting with an all initialized buffer of the CLDFBaccording to an embodiment of the present invention, the synthesisfilterbank as well as the analysis filterbank is capable of providing“meaningful” data sooner compared to the SBR QMF filterbank due to theshape of the window function. In other words, due to the shape of theanalysis or synthesis window function 700, samples processed by windowfunctions indicative of the significant contribution is sooner possible.As a consequence, the synthesis prototype or synthesis window functionof the CLDFB introduces only an overlap delay of 32 samples taking intoaccount the delay already introduced by the modulation core 750. Thefirst portion 420 or first group 420 of window coefficients of thewindow function 700 according to an embodiment of the present inventioncomprises in an embodiment according to the present invention the 96window coefficients corresponding to the delay caused by the left-sideoverlap 760 together with the modulation core delay 750.

The same delay is introduced by the analysis filterbank or the analysisprototype function. The reason is that the analysis filterbank is basedon the time-reverse version of the synthesis window function orprototype function. Thus, the overlap delay is introduced on the rightside comprising the same overlap size as for the synthesis filterbank.Hence, in the case of an original QMF prototype filterbank, also a delayof 288 samples is introduced while for an analysis filterbank accordingto an embodiment of the present invention only 32 samples are introducedas a delay.

The table shown in FIG. 14 a provides an overview of the delay withdifferent modification stages assuming a frame length of 480 samples anda sampling rate of 48 kHz. In a standard configuration comprising anAAC-LD codec along with a standard SBR tool, the MDCT and IMDCTfilterbanks in the dual-rate mode cause a delay of 40 ms. Moreover, theQMF tool itself causes a delay of 12 ms. Moreover, due to anSBR-overlap, a further delay of 8 ms is generated so that the overalldelay of this codec is in the range of 60 ms.

In comparison an AAC-ELD codec comprising low-delay versions of the MDCTand the IMDCT generate in the dual-rate approach a delay of 30 ms.Compared to the original QMF filterbank of an SBR tool, employing acomplex-valued low-delay filterbank according to an embodiment of thepresent invention will result in a delay of only 1 ms compared to 12 msof the original QMF tool. By avoiding the SBR-overlap the additionaloverlap of 8 ms of a straightforward combination of an AAC-LD and theSBR tool can be avoided completely. Therefore, the enhanced low-delayAAC codec is capable of an overall algorithmic delay of 31 ms ratherthan 60 ms for the straightforward combination previously outlined.Therefore, it can be seen that the combination of the described delayreduction methods indeed results in a total delay saving of 29 ms.

The table in FIG. 14 b gives a further overview of the overall codecdelay caused by the original and the proposed filterbank versions in asystem as shown in FIGS. 5 and 6. The data and values given in FIG. 14 bare based on a sampling rate of 48 kHz and a core coder frame size of480 samples. Due to the dual-rate approach of a SBR system as shown anddiscussed in FIGS. 5 and 6, the core coder is effectively running at asampling rate of 24 kHz. Since the framing delay of 64 samples for themodulation core is already introduced by the core coder, it can besubtracted from the standalone delay values of the two filterbanks asdescribed in the context of FIG. 13.

The table in FIG. 14 b underlines that it is possible to reduce theoverall delay of the enhanced low-delay AAC codec comprising thelow-delay versions of the MDCT and the IMDCT (LD MDCT and LD IMDCT).While an overall algorithmic delay of 42 ms is achievable only byemploying the low-delay versions of the MDCT and the IMDCT as well asthe original QMF filterbanks, by using complex-valued low-delayfilterbanks according to embodiments of the present invention instead ofthe conventional QMF filterbanks, the overall algorithmic delay can besignificantly reduced to only 31.3 ms.

To evaluate the quality of the filterbanks according to embodiments ofthe present invention and systems comprising one or more filterbanks,listening tests have been carried out, from which it can be concludedthat filterbanks according to embodiments of the present invention keepthe audio quality of AAC-ELD at the same level and do not introduce anydegradation, neither for the complex SBR mode nor for the real-valuedlow-power SBR mode. Thus, the delay-optimized filterbanks according toembodiments of the present invention do not introduce any burden on theaudio quality although they are capable of reducing the delay by morethan 10 ms. For the transient items it can even be observed that someslight, but not statistically significant improvements are achievable.The above-mentioned improvements have been observed during listeningtests of castagnettes and glockenspiels.

In order to further verify that the downsampling in the case of a32-band filterbank according to an embodiment of the present inventionworks equally well for the filterbanks according to the presentinvention compared to QMF filterbanks, the following evaluation wasperformed. First, a logarithmic sine sweep was analyzed with adownsampled 32-band filterbank, wherein the 32 upper bands, initializedwith zeros, were added. Afterwards, the outcome was synthesized by a64-band filterbank, downsampled again and compared to the originalsignal. Using a conventional SBR QMF prototype filter results in asignal-to-noise ratio (SNR) of 59.5 dB. A filterbank according to thepresent invention, however, achieves an SNR value of 78.5 dB, whichillustrates that filterbanks according to embodiments of the presentinvention also perform in the downsampled version at least as well asthe original QMF filterbanks.

In order to show that this delay-optimized, non-symmetric filterbankapproach as employed in embodiments according to the present inventiondoes provide additional value compared to a classical filterbank with asymmetric prototype, asymmetric prototypes will be compared withsymmetric prototypes having the same delay in the following.

FIG. 15 a shows a comparison of a frequency response in a far-fieldillustration of a filterbank according to the present inventionemploying a low-delay window (graph 800) compared to the frequencyresponse of a filterbank employing a sine-window having a length of 128taps (graph 810). FIG. 15 b shows a magnification of the frequencyresponse in the near-field of the same filterbanks employing the samewindow functions as outlined before.

A direct comparison of the two graphs 800, 810 shows that the frequencyresponse of the filterbank employing a low-delay filterbank according toan embodiment of the present invention is significantly better than thecorresponding frequency response of a filterbank employing a sine-windowof 128 taps having the same delay.

Also, FIG. 16 a shows a comparison of different window functions with anoverall delay of 127 samples. The filterbank (CLDFB) with 64 bandscomprises an overall delay of 127 samples including the framing delayand the overlap delay. A modulated filterbank with a symmetric prototypeand the same delay would, therefore, have a prototype of a length of128, as already illustrated in the context of FIGS. 15 a and 15 b. Forthese filterbanks with 50% overlap, such as, for instance, the MDCT,sine-windows or Kaiser-Bessel-derived windows generally provide a goodchoice for prototypes. Hence, in FIG. 16 a an overview of a frequencyresponse of a filterbank employing a low-delay window as a prototypeaccording to an embodiment of the present invention is compared to thefrequency responses of alternative symmetric prototypes with the samedelay. FIG. 16 a shows, apart from the frequency response of thefilterbank according to the present invention (graph 800) and thefrequency response of a filterbank employing a sine-window (graph 810),as already shown in FIGS. 15 a and 15 b, furthermore two KBD windowsbased on the parameters α=4 (graph 820) and α=6 (graph 830). Both, FIG.16 a and the close-up of FIG. 16 a shown in FIG. 16 b, clearly show thata much better frequency response can be achieved with a filterbankaccording to an embodiment of the present invention having anon-symmetric window function or a prototype filter function with thesame delay.

To illustrate this advantage on a more general basis, in FIG. 17 twofilterbank prototypes with delay values different from the previouslydescribed filterbank are compared. While the filterbank according to thepresent invention, which was considered in FIGS. 15 and 16, has anoverall delay of 127 samples, which corresponds to an overlap of 8blocks into the past and 0 blocks into the future (CLDFB 80), FIG. 17shows a comparison of the frequency responses of two differentfilterbank prototypes with a same delay of 383 samples. To be moreprecise, FIG. 17 shows a frequency response of a non-symmetric prototypefilterbank (graph 840) according to an embodiment of the presentinvention, which is based on an overlap of 6 blocks of time-domainsamples into the past and 2 blocks of time-domain samples into thefuture (CLDFB 62). Moreover, FIG. 17 also shows the frequency response(graph 850) of a corresponding symmetric prototype filter function alsohaving a delay of 383 samples. It can be seen that with the same delayvalue a non-symmetric prototype or window function achieves a betterfrequency response than a filterbank with a symmetric window function orprototype filter. This demonstrates the possibility of a bettertrade-off between delay and quality, as indicated earlier.

FIG. 18 illustrates the temporal masking effect of the human ear. When asound or a tone appears at a moment in time indicated by a line 860 inFIG. 18, a masking effect concerning the frequency of the tone or thesound and neighboring frequencies arises approximately 20 ms before theactual sound starts. This effect is called pre-masking and is one aspectof the psychoacoustic properties of the human ear.

In the situation illustrated in FIG. 18, the sound remains audible forapproximately 200 ms until a moment in time illustrated by a line 870.During this time, a masker of the human ear is active, which is alsocalled simultaneous masking. After the sound stops (illustrated by line870), the masking of the frequency in the neighboring frequency of thetone slowly decays over a period of time of approximately 150 ms asillustrated in FIG. 18. This psychoacoustic effect is also referred toas post-masking.

FIG. 19 illustrates a comparison of a pre-echo behavior of aconventional HE-AAC coded signal and an HE-AAC coded signal which isbased on a filterbank employing a low-delay filterbank (CMLDFB)according to an embodiment of the present invention. FIG. 19 aillustrates the original time signal of castagnettes, which have beenprocessed with a system comprising an HE-AAC codec(HE-AAC=high-efficiency advanced audio codec). The output of the systembased on the conventional HE-AAC is illustrated in FIG. 19 b. A directcomparison of the two signals, the original time signal and the outputsignal of the HE-AAC codec shows that prior to the beginning of thesound of the castagnettes in the area illustrated by an arrow 880 theoutput signal of the HE-AAC codec comprises noticeable pre-echo effects.

FIG. 19 c illustrates an output signal of a system comprising an HE-AACbased on filterbanks comprising CMLDFB-windows according to anembodiment of the present invention. The same original time signalsindicated in FIG. 19 a and processed using filterbanks according to anembodiment of the present invention show a significantly reducedappearance of pre-echo effects just prior to the beginning of acastagnettes signal as indicated by an arrow 890 in FIG. 19 c. Due tothe pre-masking effect as described in the context of FIG. 18, thepre-echo effect indicated by the arrow 890 of FIG. 19 c will be farbetter masked than the pre-echo effects indicated by the arrow 880 inthe case of the conventional HE-AAC codec. Therefore, the pre-echobehavior of filterbanks according to the present invention, which isalso a result of the significantly reduced delay compared toconventional filterbanks, causes the output to be far better fitted tothe temporal masking properties and the psychoacoustics of the humanear. As a consequence, as was already indicated when describing thelistening tests, employing filterbanks according to an embodiment of thepresent invention can even lead to an improvement concerning the qualitycaused by the reduced delay.

Embodiments according to the present invention do not increase thecomputational complexity compared to conventional filterbanks. Low-delayfilterbanks use the same filter length and the same mode of modulationas, for instance, QMF filterbanks in the case of SBR systems such thatthe computational complexity does not increase. In terms of memoryrequirements due to the asymmetric nature of the prototype filters, theROM (read-only memory) memory requirement for the synthesis filterbankincrease approximately by 320 words in the case of a filterbank based onN=64 samples per block and T=10 blocks per frame. Moreover, in the caseof an SBR-related system, the memory requirement further increases byanother 320 words if the analysis filter is stored separately.

However, as the current ROM requirements for an AAC-ELD core isapproximately 2.5 k words (kilo words) and for the SBR implementationanother 2.5 k words, the ROM requirement is only moderately increased byabout 10%. As a possible trade-off between memory and complexity, if alow memory consumption is paramount, a linear interpolation can be usedto generate the analysis filter from the synthesis filter as outlined inthe context of FIG. 3 and equation (15). This interpolation operationincreases the number of instructions by only approximately 3.6%.Therefore, a replacement of the conventional QMF filterbanks in theframework of SBR modules by: low-delay filterbanks according toembodiments of the present invention, the delay can be reduced in someembodiments by more than 10 ms without any degradation of audio qualityor noticeable increase in complexity.

Embodiments according to the present invention therefore relate to ananalysis or synthesis window or apparatus or method for windowing.Moreover, an analysis or synthesis filterbank or method of analyzing orsynthesizing a signal using a window is described. Naturally, thecomputer program implementing one of the above methods is alsodisclosed.

Implementation according to embodiments of the present invention can becarried out as hardware implementations, software implementations or acombination of both. Data, vectors and variables generated, received orotherwise stored to be processed may be stored in different kinds ofmemories such as random-access memories, buffers, Read-Only memories,non-volatile memories (e.g. EEPROMs, flash-memories) or other memoriessuch as magnetic or optical memories. A storage position may, forinstance, be one or more memory units needed to store or save therespective amounts of data, such as variables, parameters, vectors,matrices, window coefficients or other pieces of information and data.

Software implementations may be operated on different computers,computer-like systems, processors, ASICs (application-specificintegrated circuits) or other integrated circuits (ICs).

Depending on certain implementation requirements of embodiments of theinventive methods, embodiments of the inventive methods can beimplemented in hardware, software or in a combination of both. Theimplementation can be performed using a digital storage medium, inparticular a disc CD, a DVD or another disc having an electronicallyreadable control signal stored thereon which cooperates with aprogrammable computer system, processor or integrated circuit such thatan embodiment of the inventive method is performed. Generally, anembodiment of the present invention is, therefore, a computer programproduct with a program code stored on a machine-readable carrier, theprogram code being operated for performing an embodiment of theinventive methods when the computer program product runs on a computer,processor or integrated circuit. In other words, embodiments of theinventive methods are, therefore, a computer program having a programcode for performing at least one embodiment of the inventive methodswhen the computer program runs on a computer, processor or integratedcircuit.

An apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention comprisesan analysis windower (110) for windowing a frame (120) of time-domainaudio input samples being in a time sequence extending from an earlysample to a later sample using an analysis window function (190)comprising a sequence of window coefficients to obtain windowed sampled,the analysis window function (190) comprising a first group (200) ofwindow coefficients comprising a first portion of the sequence of windowcoefficients and a second group (210) of window coefficients comprisinga second portion of the sequence of window coefficients, the firstportion comprising less window coefficients than the second portion,wherein an energy value of the window coefficients in the first portionis higher than an energy value of the window coefficients of the secondportion, wherein the first group of window coefficients is used forwindowing later time-domain samples and the second group of windowcoefficients is used for windowing earlier time-domain samples, and acalculator (170) for calculating the audio subband values using thewindowed samples.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the analysis windowfunction (190) is asymmetric with respect to the sequence of windowcoefficients.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that an energy value of thewindow coefficients of the first portion is equal to or greater than ⅔of an energy value of all window coefficients of the sequence of windowcoefficients and an energy value of the window coefficients of thesecond portion of window coefficients is smaller than or equal to ⅓ ofan energy value of all window coefficients of the sequence of windowcoefficients.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the first portion of windowcoefficients comprises ⅓ or less than ⅓ of a total number of windowcoefficients of the sequence of window coefficients and the secondportion comprises ⅔ or more than ⅔ of the total number of windowcoefficients of the sequence of window coefficients.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that a center point of thewindow coefficients of the analysis window function (190) corresponds toa real value in an index range of the first portion of windowcoefficients.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the analysis windowfunction (190) comprises a strictly monotonic decreasing from the windowcoefficient comprising the highest absolute value of all windowcoefficients of the analysis window function (190) to a windowcoefficient of the sequence of window coefficients used for windowingthe latest time-domain audio sample.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the analysis windowfunction (190) comprises an oscillating behavior.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the window coefficientcorresponding to an index n=(T−1)·N comprises an absolute value in therange of 0.9 to 1.1, wherein an index of the sequence of windowcoefficients is an integer in the range of 0 to N·T−1, wherein thewindow coefficient used for windowing the latest time-domain audio inputsample of the frame 120 is the window coefficient corresponding to theindex N·T−1, wherein the analysis windower (110) is adapted such thatthe frame (120) of time-domain audio input samples comprises a sequenceof T blocks (130) of time-domain audio input samples extending from theearliest to the latest time-domain audio input samples of the frame(120), each block comprising N time-domain audio input samples, andwherein T and N are positive integers and T is larger than 4.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the window coefficientcorresponding to the index of the window coefficients n=N·T−1 comprisesan absolute value of less than 0.02.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis window (110) is adapted such that windowing comprisesmultiplying the time-domain audio input samples x(n) of the frame (120)to obtain the windowed samples z(n) of the windowed frame based on theequationz(n)=x(n)·c(n)wherein n is an integer indicating an index of the sequence of windowcoefficients in the range of 0 to T·N−1, wherein c(n) is the windowcoefficient of the analysis window function corresponding to the indexn, wherein x(N·T−1) is the latest time-domain audio input sample of aframe (120) of time-domain audio input samples, wherein the analysiswindower (110) is adapted such that the frame (120) of time-domain audioinput samples comprises a sequence of T blocks (130) of time-domainaudio input samples extending from the earliest to the latesttime-domain audio input samples of the frame (120), each blockcomprising N time-domain audio input samples, and wherein T and N arepositive integers and T is larger than 4.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the window coefficientsc(n) obey the relations given in the table in Annex 4.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theapparatus (100) is adapted to using an analysis window function (190)being a time-reversed or index-reversed version of a synthesis windowfunction (370) to be used for the audio subband values.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the first portion of theanalysis window function comprises a window coefficient having anabsolute maximum value being greater than 1.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that all window coefficients ofthe sequence of window coefficients are real-valued window coefficients.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the frame (120) oftime-domain audio input samples comprises a sequence of T blocks (130)of time-domain audio input samples extending from the earliest to thelatest time-domain audio input samples of the frame (120), each blockcomprising N time-domain audio input samples, wherein T and N arepositive integers and T is larger than 4.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that windowing comprises anelement-wise multiplying of the time-domain audio input samples of theframe (120) with the window coefficients of the sequence of windowcoefficients.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that each time-domain audioinput sample is element-wise multiplied with a window coefficient of theanalysis window function according to a sequence of time-domain audioinput samples and the sequence of window coefficients.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that for each time-domain audioinput sample of the frame (120) of time-domain audio input samplesexactly one windowed sample is generated.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the window coefficientcorresponding to an index of the window coefficients n=(T−3)·N comprisesa value of less than −0.1, wherein the index of the sequence of windowcoefficients is an integer in the range of 0 to N·T−1, and wherein thewindow coefficient used for windowing the latest time-domain audio inputsample is the window coefficient corresponding to the index N·T−1.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the first portion of windowcoefficients comprises 3/2·N window coefficients and the second portionof window coefficients comprises (T−3/2)·N window coefficients of thesequence of window coefficients.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the window coefficientsc(n) fulfill the relations given in the table in Annex 3.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the window coefficientsc(n) fulfill the relations given in the table in Annex 2.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theanalysis windower (110) is adapted such that the window coefficientsc(n) comprise the values given in the table in Annex 1.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theapparatus (100) is adapted such that the present frame (120) oftime-domain audio input samples to be processed is generated by shifting(T−1) later blocks of a directly preceding frame (120) of time-domainaudio input samples by one block towards the earlier time-domain audioinput samples and by adding one block (220) of fresh time-domain audiosamples as the block comprising the latest time-domain audio inputsamples of the present frame (120).

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theapparatus (100) is adapted such that the present frame (120) oftime-domain audio input samples x(n) to be processed is generated basedon shifting the time-domain audio input samples x_(prev)(n) of thedirectly preceding frame (120) of time-domain audio input samples basedon the equationx(n−32)=x _(prev)(n)for a time or sample index n=32, . . . , 319, and wherein the apparatus(100) is further adapted to generating the time-domain audio inputsamples x(n) of the present frame (120) of time-domain audio inputsamples by including 32 next incoming time-domain input samplesaccording to an order of the incoming time-domain audio input samples ofdecreasing time or sample indices n for the time-domain audio inputsamples x(n) of the present frame (120) starting at the time or sampleindex n=31.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, thecalculator (170) comprises a time/frequency converter adapted togenerating the audio subband values such that all subband values basedon one frame (150) of windowed samples represent a spectralrepresentation of the windowed samples of the frame (150) of windowedsamples.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, thetime/frequency converter is adapted to generating complex-valued orreal-valued audio subband values.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, thecalculator (170) is adapted to calculating one audio subband value foreach time-domain audio input sample of one block (130) of time-domainaudio input samples, wherein calculating each audio subband value oreach of the time-domain audio input samples of one block (130) oftime-domain audio input samples is based on the windowed samples of thewindowed frame (150).

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, thecalculator (170) is adapted to calculating the audio subband valuesbased on multiplying the windowed samples (150) with a harmonicallyoscillating function for each subband value and summing up themultiplied windowed samples, wherein a frequency of the harmonicallyoscillating function is based on a center frequency of a correspondingsubband of the subband values.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, thecalculator (170) is adapted such that the harmonically oscillatingfunction is a complex exponential function, a sine function or a cosinefunction.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, thecalculator (170) is adapted to calculating the audio subband valuesw_(kl) based on the equation

$u_{n} = {\sum\limits_{j = 0}^{4}{z\left( {n + {j \cdot 64}} \right)}}$for n=0, . . . , 63 and

$w_{k\; 1} = {\sum\limits_{n = 0}^{63}{u_{n} \cdot 2 \cdot {f_{osc}\left( {\frac{\pi}{64} \cdot \left( {k + 0.5} \right) \cdot \left( {{2\; n} - 95} \right)} \right)}}}$for k=0, . . . , 31, wherein z(n) is a windowed sample corresponding toan index n, wherein k is a subband index, wherein 1 is an index of ablock (180) of audio subband values and wherein f_(osc)(x) is anoscillating function depending on a real-valued variable x.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, thecalculator (170) is adapted such that the oscillating functionf_(osc)(x) isf _(osc)(x)=exp(i·x)orf _(osc)(x)=cos(x)orf _(osc)(x)=sin(x)wherein i is the imaginary unit.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theapparatus (100) is adapted to processing a frame (120) of real-valuedtime-domain audio input samples.

In an apparatus for generating audio subband values in audio subbandchannels according to an embodiments of the present invention, theapparatus (100) is adapted to providing a signal indicative of asynthesis window function (370) to be used with the audio subband valuesor indicative of the analysis window function (190) used for generatingthe audio subband values.

An apparatus for generating time-domain audio samples according to anembodiments of the present invention comprises a calculator (310) forcalculating a sequence (330) of intermediate time-domain samples fromaudio subband values in audio subband channels, the sequence comprisingearlier intermediate time-domain samples and later time-domain samples,a synthesis windower (360) for windowing the sequence (330) ofintermediate time-domain samples using a synthesis window function (370)comprising a sequence of window coefficients to obtain windowedintermediate time-domain samples, the synthesis window function (370)comprising a first group (420) of window coefficients comprising a firstportion of the sequence of window coefficients and a second group (430)of window coefficients comprising a second portion of the sequence ofwindow coefficients, the first portion comprising less windowcoefficients than the second portion, wherein an energy value of thewindow coefficients in the first portion is higher than an energy valueof the window coefficients of the second portion, wherein the firstgroup of window coefficients is used for windowing later intermediatetime-domain samples and the second group of window coefficients is usedfor windowing earlier intermediate time-domain samples, and anoverlap-adder output stage (400) for processing the windowedintermediate time-domain samples to obtain the time-domain samples.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that an energy value of the window coefficients of thefirst portion of window coefficients is larger than or equal to ⅔ of anenergy value of all window coefficients of the synthesis window function(370) and an energy value of the second portion of window coefficientsis smaller than, or equal to ⅓ of the energy value of all windowcoefficients of the synthesis window function.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the first portion of window coefficients comprises ⅓or less than ⅓ of the total number of all window coefficients of thesequence of window coefficients and the second portion of windowcoefficients comprises ⅔ or more than ⅔ of the total number of windowcoefficients of the sequence of window coefficients.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that a center point of the window coefficients of thesynthesis window function (370) corresponds to a real value in an indexrange of the first portion of window coefficients.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the synthesis window function comprises a strictlymonotonic increase from the window coefficient of the sequence of windowcoefficients used for windowing the latest intermediate time-domainsample to the window coefficient comprising the highest absolute valueof all window coefficients of the synthesis window function.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the synthesis window function (370) comprises anoscillating behavior.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the window coefficientcorresponding to an index n=N comprises an absolute value in the rangebetween 0.9 and 1.1, wherein the index n of the sequence of windowcoefficients is an integer in the range of 0 to T·N−1, wherein thewindow coefficient used for windowing the latest intermediatetime-domain sample is the window coefficient corresponding to the indexn=0, wherein T is an integer larger than 4 indicating the number ofblocks comprises in the frame (330) of intermediate time-domain samples,wherein the apparatus (300) is adapted to generating a block (410) oftime-domain audio samples, the block (410) of time-domain audio samplescomprising N time-domain audio samples, wherein N is a positive integer.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the window coefficient corresponding to the index n=0comprises an absolute value smaller than or equal to 0.02.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the window coefficient corresponding to an index n=3Nis smaller than −0.1, wherein the apparatus (300) is adapted togenerating a block (410) of time-domain audio samples, the block (410)of time-domain audio samples comprising N time-domain audio samples,wherein N is a positive integer.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis window (360) isadapted such that the windowing comprises multiplying the intermediatetime-domain samples g(n) of the sequence of intermediate time-domainsamples to obtain the windowed samples z(n) of the windowed frame (380)based on the equationz(n)=g(n)·c(T·N−1−n)for n=0, . . . , T·N−1.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the window coefficient c(n) fulfill the relationsgiven in the table in Annex 4.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the apparatus (300) is adapted tousing the synthesis window function (370) being a time-reverse orindex-reversed version of an analysis window function (190) used forgenerating the audio subband values.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the apparatus (300) is adapted togenerating a block (410) of time-domain audio samples, the block (410)of time-domain audio samples comprising N time-domain audio samples,wherein N is a positive integer.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the apparatus (300) is adapted togenerating the block (410) of time-domain audio samples, based on ablock (320) of audio subband values comprising N audio subband valuesand wherein the calculator (310) is adapted to calculating the sequence(330) of intermediate time-domain audio samples comprising T·Nintermediate time-domain audio samples, wherein T is a positive integer.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the synthesis window function is asymmetric withrespect to the sequence window coefficient.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the first portion comprises a maximum value of allwindow coefficients of the synthesis window function having an absolutevalue larger than 1.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the first portion comprises 3/2·N window coefficientsand the second portion of window coefficients comprises (T−3/2)·N windowcoefficients, wherein T is an index greater or equal to 4 indicating anumber of blocks 340 comprised in the frame (330) of intermediatetime-domain samples.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that windowing the sequence of intermediate time-domainsamples comprises an element-wise multiplying of the intermediatetime-domain samples with a window coefficient.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that each intermediate time-domain sample is element-wisemultiplied with the window coefficient of the synthesis window function(370) according to the sequence of intermediate time-domain samples andthe sequence of window coefficients.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis window (360) isadapted such that the window coefficients of the synthesis windowfunction (370) are real-valued values.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the window coefficient c(n) fulfill the relationsgiven in the table in Annex 3.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the window coefficients c(n) fulfill the relationsgiven in the table in Annex 2.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the synthesis windower (360) isadapted such that the window coefficients c(n) comprise the values givenin the table in Annex 1.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the calculator (310) is adapted tocalculating the intermediate time-domain samples of the sequence ofintermediate time-domain samples based on multiplying the audio subbandvalues with a harmonic oscillating function and summing up themultiplied audio subband values, wherein the frequency of theharmonically oscillating function is based on a center frequency of thecorresponding subband.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the calculator (310) is adaptedsuch that the harmonically oscillating function is a complex exponentialfunction, a sine-function or a cosine-function.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the calculator (310) is adapted tocalculating real-valued intermediate time-domain samples based on thecomplex-valued or real-valued audio subband values.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the calculator (310) is adapted tocalculating the sequence of real-valued intermediate time-domain samplesz(i,n) based on the equation

$z_{i,n} = {{- \frac{1}{N}}{\sum\limits_{k = 0}^{N - 1}{{Re}\left( {X_{i,k} \cdot {f_{osc}\left( {\frac{\pi}{N}{\left( {n + \frac{1}{2} - \frac{N}{2}} \right) \cdot \left( {k + \frac{1}{2}} \right)}} \right)}} \right)}}}$for an integer n in the range of 0 to N·T−1, wherein Re(x) is the realpart of the complex-valued number x, π=3.14 . . . is the circular numberand f_(osc)(x) is a harmonically oscillating function, whereinf _(osc)(x)=exp(i·x),when the audio subband values provided to the calculator are complexvalues, wherein I is the imaginary unit, and whereinf _(osc)(x)=cos(x)when the audio subband values provided to the calculator (310) are realvalues.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the calculator (310) comprises afrequency/time converter adapted to generating the sequence ofintermediate time-domain samples, such that the audio subband valuesprovided to the calculator (310) represent a spectral representation ofthe sequence of intermediate time-domain samples.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the frequency/time converter isadapted to generating the sequence of intermediate time/domain samplesbased on complex-valued or real-valued audio subband values.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the calculator (310) is adapted tocalculating the sequence of intermediate time-domain samples g(n) fromthe audio subband values X(k) based on the equationv(n)=v _(prev)(n−2N)for an integer n in the range of 20N−1 and 2N,

${v(n)} = {\sum\limits_{k = 0}^{N - 1}{{Re}\left( {{{X(k)} \cdot \frac{1}{64}}{\exp\left( {{\mathbb{i}}\frac{\pi}{2\; N}{\left( {k + \frac{1}{2}} \right) \cdot \left( {{2\; n} - \left( {N - 1} \right)} \right)}} \right)}} \right)}}$for the integer n in the range of 0 and 2N−1 andg(2N·j+k)=v(4Nj+k)g(2N·j+N+k)=v(4Nj+3N+k)for an integer j in the range of 0 and 4 and for an integer k in therange of 0 and N−1, wherein N is an integer indicating the number ofaudio subband values and the number of the time-domain audio samples,wherein v is a real-valued vector, wherein v_(prev) is a real-valuedvector v of the directly previous generation of time-domain audiosamples, wherein i is the imaginary unit and π is the circular number.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the calculator (310) is adapted tocalculating the sequence of intermediate time-domain samples g(n) fromthe audio subband values X(k) based on the equationv(n)=v _(prev)(n−2N)for an integer n in the range of 20N−1 and 2N,

${v(n)} = {\sum\limits_{k = 0}^{N - 1}{{X(k)} \cdot \frac{1}{32} \cdot {\cos\left( {\frac{\pi}{2\; N}{\left( {k + \frac{1}{2}} \right) \cdot \left( {{2\; n} - \left( {N - 1} \right)} \right)}} \right)}}}$for the integer n in the range of 0 and 2N−1 andg(2N·j+k)=v(4Nj+k)g(2N·j+N+k)=v(4Nj+3N+k)for an integer j in the range of 0 and 4 and for an integer k in therange of 0 and N−1, wherein N is an integer indicating the number ofaudio subband values and the number of the time-domain audio samples,wherein v is a real-valued vector, wherein v_(prev) is a real-valuedvector v of the directly previous generation of time-domain audiosamples and wherein π is the circular number.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the overlap-add output stage (400)is adapted to processing the windowed intermediate time-domain samplesin an overlapping manner, based on T consecutively provided blocks (320)of audio subband values.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the overlap-add output stage (400)is adapted to providing the time-domain samples out_(l)(n), wherein n isan integer indicating a sample index based on the equation

${{{out}_{1}(n)} \cdot} = {\sum\limits_{k = 0}^{T - 1}z_{{({1 - k})},{n + {k \cdot N}}}}$wherein z_(l,n) is a windowed intermediate time-domain samplecorresponding to a sample index n and a frame or sequence index l in therange from 0 to T−1, wherein l=0 corresponds to the latest frame orsequence and smaller values of l to previously generated frames orsequences.

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the overlap-add output stage (400)is adapted to providing the time-domain samples out(k) based on theequation

${{{out}(k)} = {\sum\limits_{k = 0}^{9}{w\left( {{N \cdot n} + k} \right)}}},$wherein w is a vector comprising the windowed intermediate time-domainsamples and k is an integer indicating an index in the range between 0and (N−1).

In an apparatus for generating time-domain audio samples according to anembodiments of the present invention, the apparatus (300) is adapted toreceiving a signal indicative of the analysis window function (190) usedfor generating the audio subband values, or indicative of the synthesiswindow function (370) to be used for generating the time-domain audiosamples.

According to an embodiments of the present invention, an encoder (510)comprises an apparatus (560) for generating audio subband values inaudio subband channels according to according to an embodiment of thepresent invention.

According to an embodiments of the present invention, an encoder (510)further comprises a quantizer and coder (570) coupled to the apparatus(560) for generating audio subband values and adapted to quantizing andcoding the audio subband values output by the apparatus (560) andoutputting the quantized coded audio subband values.

According to an embodiments of the present invention, a decoder (580)comprises an apparatus (600) for generating time-domain audio samplesaccording to an embodiment of the present invention.

According to an embodiments of the present invention, a decoder (580)further comprises a decoder and dequantizer (590) adapted to receivingcoded and quantized audio subband values, coupled to the apparatus (600)for generating time-domain audio samples and adapted to providing thedecoded and dequantized audio subband values as the audio subband valuesto the apparatus (600).

According to an embodiments of the present invention, a SBR encoder(520) comprises an apparatus (530) for generating audio subband valuesin audio subband channels, based on a frame of time-domain audio inputsamples provided to the SBR encoder (520) and a SBR parameter extractionmodule (540) coupled to the apparatus (530) for generating audio subbandvalues and adapted to extracting and outputting SBR parameters based onthe audio subband values.

According to an embodiments of the present invention, a system (610)comprises an apparatus (620) for generating audio subband values from aframe of time-domain audio input samples provided to the system (610);and an apparatus (640) for generating time-domain audio samples based onthe audio subband values generated by the apparatus (640) for generatingaudio subband values.

According to an embodiments of the present invention, a system (610) isa SBR decoder.

According to an embodiments of the present invention, a system furthercomprises a HF-generator (630) interconnected between the apparatus(620) for generating audio subband values and the apparatus (640) forgenerating time-domain audio samples and adapted to receiving SBR dataadapted to modifying or adding audio subband values based on the SBRdata and the audio subband values from the apparatus (620) forgenerating audio subband values.

With respect to all apparatuses and methods according to embodiments ofthe present invention, depending on implementational details, to obtainthe window coefficients based on the values given in the tables in theAnnexes 1 and 3, an additional sign changes with respect to the windowcoefficients corresponding to the indices 128 to 255 and 384 to 511(multiplication with factor (−1)) can be implemented. In other words,the window coefficients of the window function are based on the windowcoefficients given in table in Annex 1. To obtain the windowcoefficients of the window function shown in the figures, the windowcoefficients in the table corresponding to the indices 0 to 127, 256 to383 and 512 to 639 have to be multiplied by (+1) (i.e. no sign change)and the window coefficients corresponding to the indices 128 to 255 and384 to 511 have to be multiplied by (−1) (i.e. a sign change) to obtainthe window coefficients of the window function shown. Accordingly therelations given in the table in Annex 3 have to be treated accordingly.

It should be noted, that in the framework of the present applicationunder an equation being based on an equation an introduction ofadditional delays, factors, additional coefficients and an introductionof another simple function is understood. Further, simple constants,constant addends etc., can be dropped. Moreover, algebraictransformations, equivalence transformations and approximations (e.g. aTaylor approximation) not changing the result of the equation at all orin a significant manner are also included. In other words, both slightmodifications as well as transformations leading to essentially in termsof the result identical are included in the case that an equation orexpression is based on an equation or expression.

While this invention has been described in terms of several embodiments,there are alterations, permutations, and equivalents which fall withinthe scope of this invention. It should also be noted that there are manyalternative ways of implementing the methods and compositions of thepresent invention. It is therefore intended that the following appendedclaims be interpreted as including all such alterations, permutationsand equivalents as fall within the true spirit and scope of the presentinvention.

Annex 1 w[0] = 1.129580193872797e−002 w[1] = 2.353059744904218e−002 w[2]= 3.450718748721251e−002 w[3] = 4.634695977000525e−002 w[4] =5.918677345174197e−002 w[5] = 7.325978412117062e−002 w[6] =8.829745229234007e−002 w[7] = 1.042033024802571e−001 w[8] =1.206924277410051e−001 w[9] = 1.376149808913910e−001 w[10] =1.547461142258783e−001 w[11] = 1.719726384566089e−001 w[12] =1.891590407342011e−001 w[13] = 2.062605107774960e−001 w[14] =2.232276864673650e−001 w[15] = 2.400768261284114e−001 w[16] =2.568176309566753e−001 w[17] = 2.734977190313227e−001 w[18] =2.901491317310591e−001 w[19] = 3.068186515423912e−001 w[20] =3.235298682841570e−001 w[21] = 3.403074146062977e−001 w[22] =3.571527896130669e−001 w[23] = 3.740643974275026e−001 w[24] =3.910243970160607e−001 w[25] = 4.080154903861317e−001 w[26] =4.250144186334534e−001 w[27] = 4.420013942269341e−001 w[28] =4,589582896478246e−001 w[29] = 4.758753745532750e−001 w[30] =4.927463828072591e−001 w[31] = 5.095720854151864e−001 w[32] =5.263554446856779e−001 w[33] = 5.430990601899994e−001 w[34] =5.598052330684253e−001 w[35] = 5.764734796907189e−001 w[36] =5.930981800982896e−001 w[37] = 6.096690552916387e−001 w[38] =6.261725236758639e−001 w[39] = 6.425939632009995e−001 w[40] =6.589148753746076e−001 w[41] = 6.751199626157149e−001 w[42] =6.911981575264606e−001 w[43] = 7.071447728928043e−001 w[44] =7.229599104052475e−001 w[45] = 7:386515025302785e−001 w[46] =7.542294504292890e−001 w[47] = 7.697093346240386e−001 w[48] =7.851012620144958e−001 w[49] = 8.004165237845137e−001 w[50] =8.156523162880560e−001 w[51] = 8.308039608112368e−001 w[52] =8.458450064727010e−001 w[53] = 8.607492455327098e−001 w[54] =8.754640719350776e−001 w[55] = 8.899474405744183e−001 w[56] =9.041286138017367e−001 w[57] = 9.179666107725365e−001 w[58] =9.313874086278087e−001 w[59] = 9.443802853939540e−001 w[60] =9.568885413848645e−001 w[61] = 9.690016637782843e−001 w[62] =9.807691702375303e−001 w[63] = 9.927543720639498e−001 w[64] =1.001463112557766e+000 w[65] = 1.006893331637123e+000 w[66] =1.012508393574432e+000 w[67] = 1.017729040219375e+000 w[68] =1.022470190536100e+000 w[69] = 1.026615653698808e+000 w[70] =1.030198648769593e+000 w[71] = 1.033205850580933e+000 w[72] =1.035694432087486e+000 w[73] = 1.037683165297586e+000 w[74] =1.039227995800217e+000 w[75] = 1.040349586463588e+000 w[76] =1.041086497214721e+000 w[77] = 1.041443375950143e+000 w[78] =1.041434355650865e+000 w[79] = 1.041043184216171e+000 w[80] =1.040262316588456e+000 w[81] = 1.039061496136853e+000 w[82] =1.037422300157921e+000 w[83] = 1.035311720204252e+000 w[84] =1.032712952177121e+000 w[85] = 1.029600494883906e+000 w[86] =1.025966756910904e+000 w[87] = 1.021798805583990e+000 w[88] =1.017100128250049e+000 w[89] = 1.011867706519706e+000 w[90] =1.006109248754940e+000 w[91] = 9.998285752401580e−001 w[92] =9.930379854679836e−001 w[93] = 9.857387823493258e−001 w[94] =9.779405164766706e−001 w[95] = 9.696426101291272e−001 w[96] =9.608519516143015e−001 w[97] = 9.515674613550604e−001 w[98] =9.417975696327747e−001 w[99] = 9.315442093447622e−001 w[100] =9.208194746232827e−001 w[101] = 9.096310803629866e−001 w[102] =8.979959173503500e−001 w[103] = 8.859232320517536e−001 w[104] =8.734366852542127e−001 w[105] = 8.605542791988831e−001 w[106] =8.472987145504696e−001 w[107] = 8.336863467961255e−001 w[108] =8.197387292306723e−001 w[109] = 8.054701312929008e−001 w[110] =7.908995350037713e−001 w[111] = 7.760385598209244e−001 w[112] =7.609051036128973e−001 w[113] = 7.455111681431031e−001 w[114] =7.298745530879272e−001 w[115] = 7.140087729493950e−001 w[116] =6.979336851549095e−001 w[117] = 6.816667882498023e−001 w[118] =6.652304141388827e−001 w[119] = 6.486437667370537e−001 w[120] =6.319284031798550e−001 w[121] = 6.151031151692835e−001 w[122] =5.981877665956570e−001 w[123] = 5.811992722116214e−001 w[124] =5.641522833259215e−001 w[125] = 5.470652177576862e−001 w[126] =5.299509559653194e−001 w[127] = 5.128557121424191e−001 w[128] =−4.956175421414453e−001 w[129] = −4.782650346610896e−001 w[130] =−4.609828932783459e−001 w[131] = −4.437530233023859e−001 w[132] =−4.265950246465440e−001 w[133] = −4.095160467543179e−001 w[134] =−3.925409172155113e−001 w[135] = −3.756821671788237e−001 w[136] =−3.589626517817934e−001 w[137] = −3.423942311297658e−001 w[138] =−3.259993851088293e−001 w[139] = −3.097861805973821e−001 w[140] =−2.937724988593393e−001 w[141] = −2.779637821990255e−001 w[142] =−2.623749159488041e−001 w[143] = −2.470098299603623e−001 w[144] =−2.318815478758375e−001 w[145] = −2.169925682529340e−001 w[146] =−2.023548005388463e−001 w[147] = −1.879711746686855e−001 w[148] =−1.738542127021508e−001 w[149] = −1.600061812296078e−001 w[150] =−1.464389150679625e−001 w[151] = −1.331544923127771e−001 w[152] =−1.201628679722633e−001 w[153] = −1.074630704470568e−001 w[154] =−9.506966959632511e−002 w[155] = −8.298103104739203e−002 w[156] =−7.120356992726613e−002 w[157] = −5.973741829536090e−002 w[158] =−4.859005767016811e−002 w[159] = −3.775928110298274e−002 w[160] =−2.726484300186575e−002 w[161] = −1.711323992709580e−002 w[162] =−7.298197371320593e−003 w[163] = 2.184256929356781e−003 w[164] =1.132324047372148e−002 w[165] = 2.012236990754980e−002 w[166] =2.857528272530154e−002 w[167] = 3.666942822678171e−002 w[168] =4.439683978044157e−002 w[169] = 5.177964768870787e−002 w[170] =5.881296711410786e−002 w[171] = 6.550209046893848e−002 w[172] =7.184073822817207e−002 w[173] = 7.783299328224960e−002 w[174] =8.347150698567406e−002 w[175] = 8.875756217893037e−002 w[176] =9.368651761350569e−002 w[177] = 9.826251129465624e−002 w[178] =1.024804711677230e−001 w[179] = 1.063454554357498e−001 w[180] =1.098551252869576e−001 w[181] = 1.130180022553412e−001 w[182] =1.158358935177899e−001 w[183] = 1.183233335449968e−001 w[184] =1.204854506722672e−001 w[185] = 1.223371395264402e−001 w[186] =1.238868653862843e−001 w[187] = 1.251477258491527e−001 w[188] =1.261262023246478e−001 w[189] = 1.268280540744526e−001 w[190] =1.272498700590511e−001 w[191] = 1.273590703506806e−001 w[192] =1.274567595465545e−001 w[193] = 1.275561350483646e−001 w[194] =1.273648326872248e−001 w[195] = 1.269415772180714e−001 w[196] =1.262995646340671e−001 w[197] = 1.254605188749804e−001 w[198] =1.244269583009826e−001 w[199] = 1.232131583108813e−001 w[200] =1.218183974842866e−001 w[201] = 1.202545652840080e−001 w[202] =1.185243106889108e−001 w[203] = 1.166399102636992e−001 w[204] =1.146042249339280e−001 w[205] = 1.124296184976912e−001 w[206] =1.101215600923314e−001 w[207] = 1.076972053405737e−001 w[208] =1.051641975499523e−001 w[209] = 1.025397604985405e−001 w[210] =9.982957934346254e−002 w[211] = 9.705239536075722e−002 w[212] =9.421624116597689e−002 w[213] = 9.133590931873967e−002 w[214] =8.841813387276727e−002 w[215] = 8.547715661443602e−002 w[216] =8.251962055343706e−002 w[217] = 7.955570759229536e−002 w[218] =7.657649751612349e−002 w[219] = 7.360559211914287e−002 w[220] =7.064948295960993e−002 w[221] = 6.771675107480543e−002 w[222] =6.480448458935215e−002 w[223] = 6.192692754258131e−002 w[224] =5.911363249658311e−002 w[225] = 5.637219228757212e−002 w[226] =5.368313072045600e−002 w[227] = 5.105620793438655e−002 w[228] =4.849284995895640e−002 w[229] = 4.599068181839981e−002 w[230] =4.355568588898841e−002 w[231] = 4.125570251909672e−002 w[232] =3.907137550527191e−002 w[233] = 3.696342556744636e−002 w[234] =3.493300140502248e−002 w[235] = 3.298151059524886e−002 w[236] =3.110861245410919e−002 w[237] = 2.931525594774175e−002 w[238] =2.760090729801069e−002 w[239] = 2.597956638848436e−002 w[240] =2.443433592149451e−002 w[241] = 2.296470793543091e−002 w[242] =2.156304510969632e−002 w[243] = 2.023524610221679e−002 w[244] =1.897505817503749e−002 w[245] = 1.778248750467421e−002 w[246] =1.665187994388476e−002 w[247] = 1.557759513377242e−002 w[248] =1.456208586604537e−002 w[249] = 1.361072086117313e−002 w[250] =1.270747042064656e−002 w[251] = 1.186210743261470e−002 w[252] =1.106958962776399e−002 w[253] = 1.033126278863177e−002 w[254] =9.640298325700842e−003 w[255] = 8.996371481700806e−003 w[256] =−8.407748878436545e−003 w[257] = −7.876393114319395e−003 w[258] =−7.380543918629573e−003 w[259] = −6.925141135202262e−003 w[260] =−6.500502521462604e−003 w[261] = −6.109178606718115e−003 w[262] =−5.741103163221257e−003 w[263] = −5.394569608919965e−003 w[264] =−5.063851046064050e−003 w[265] = −4.754191853611012e−003 w[266] =−4.448993249380505e−003 w[267] = −4.133639756278191e−003 w[268] =−3.811612348723333e−003 w[269] = −3.505531318950422e−003 w[270] =−3.209092846617964e−003 w[271] = −2.927159436740159e−003 w[272] =−2.653818578698405e−003 w[273] = −2.396404013961463e−003 w[274] =−2.152379960589273e−003 w[275] = −1.924844672908215e−003 w[276] =−1.699160580023900e−003 w[277] = −1.480542563288228e−003 w[278] =−1.283280633901446e−003 w[279] = −1.131859661378862e−003 w[280] =−9.730460256556873e−004 w[281] = −7.677634115875747e−004 w[282] =−5.599347984905645e−004 w[283] = −3.337966579125254e−004 w[284] =−9.099722643476421e−005 w[285] = 1.498231621816041e−004 w[286] =4.366447012116811e−004 w[287] = 6.307841647560053e−004 w[288] =6.150316826138937e−004 w[289] = 8.990255827053560e−004 w[290] =1.232134364570107e−003 w[291] = 1.471167206249042e−003 w[292] =1.697652664777771e−003 w[293] = 1.985825255428654e−003 w[294] =2.172866052963961e−003 w[295] = 1.812176023993582e−003 w[296] =1.344657262814793e−003 w[297] = 9.373975348172919e−004 w[298] =5.621720998949145e−004 w[299] = 2,048498552413189e−004 w[300] =−2.004822830002534e−004 w[301] = −6.169854804735951e−004 w[302] =−1.061498982103114e−003 w[303] = −1.594860949611097e−003 w[304] =−2.124647831574725e−003 w[305] = −2.621537051750861e−003 w[306] =−3.064311083207632e−003 w[307] = −3.460362845825662e−003 w[308] =−3.794425324215804e−003 w[309] = −4.091032597247918e−003 w[310] =−4.369553676668050e−003 w[311] = −4.554811297024067e−003 w[312] =−4.663276675479689e−003 w[313] = −4.722567636185647e−003 w[314] =−4.704321497976561e−003 w[315] = −4.636227793039124e−003 w[316] =−4.517190210387324e−003 w[317] = −4.351667566540186e−003 w[318] =−4.135130493071822e−003 w[319] = −3.870851645947402e−003 w[320] =−3.597475533950260e−003 w[321] = −3.318857985461042e−003 w[322] =−3.000422543655664e−003 w[323] = −2.658042081080524e−003 w[324] =−2.292813563887493e−003 w[325] = −1.914114740669928e−003 w[326] =−1.525818616748839e−003 w[327] = −1.156680209049319e−003 w[328] =−7.804546272743493e−004 w[329] = −4.268574601396473e−004 w[330] =−1.324291707264515e−004 w[331] = 1.218226450050751e−004 w[332] =3.189336138130849e−004 w[333] = 4.749931197951235e−004 w[334] =5.970696819774243e−004 w[335] = 6.673250213055329e−004 w[336] =6.887783835812338e−004 w[337] = 6.766320515830324e−004 w[338] =6.944123176012471e−004 w[339] = 7.139919634325070e−004 w[340] =7.154123487609100e−004 w[341] = 7.376101027486600e−004 w[342] =6.976561203768226e−004 w[343] = 5.721223454434728e−004 w[344] =2.934875643581191e−004 w[345] = 1.092526149391273e−004 w[346] =6.415402443848103e−004 w[347] = 1.194730618383423e−003 w[348] =1.557112059887280e−003 w[349] = 1.891971801393744e−003 w[350] =2.225524159129023e−003 w[351] = 2.530906981099261e−003 w[352] =2.719749515067397e−003 w[353] = 2.729136737522100e−003 w[354] =2.703019498899013e−003 w[355] = 2.630471852319136e−003 w[356] =2.470456304276468e−003 w[357] = 2.239142906871446e−003 w[358] =2.033465291493264e−003 w[359] = 1.948069005335563e−003 w[360] =1.725029670030533e−003 w[361] = 1.417366709895927e−003 w[362] =1.127141815310061e−003 w[363] = 8.089811988213151e−004 w[364] =4.708009521678285e−004 w[365] = 7.882620739833088e−005 w[366] =−2.998739993995956e−004 w[367] = −4.733148292475610e−004 w[368] =−5.791145447913150e−004 w[369] = −6.754935404082003e−004 w[370] =−8.029620210721900e−004 w[371] = −9.726698841994444e−004 w[372] =−1.196637962311630e−003 w[373] = −1.292865844760059e−003 w[374] =−1.146268465739874e−003 w[375] = −1.040598055074471e−003 w[376] =−9.767709065548874e−004 w[377] = −9.294665200453614e−004 w[378] =−9.862027119530482e−004 w[379] = −1.047654674829846e−003 w[380] =−1.099000599887377e−003 w[381] = −1.151795860160292e−003 w[382] =−1.194743370333155e−003 w[383] = −1.250742797799558e−003 w[384] =1.287819050086379e−003 w[385] = 1.263569296641556e−003 w[386] =1.226113111394085e−003 w[387] = 1.177515087338257e−003 w[388] =1.122503050159859e−003 w[389] = 1.089428846944533e−003 w[390] =1.054963366189962e−003 w[391] = 9.019128558297515e−004 w[392] =7.847839620863715e−004 w[393] = 6.205675927856794e−004 w[394] =3.157663628445906e−004 w[395] = 2.556449844935384e−004 w[396] =2.520606580606257e−004 w[397] = 2.346980949474655e−004 w[398] =2.060394037017961e−004 w[399] = 1.635905995590986e−004 w[400] =1.176237128375623e−004 w[401] = 6.193369904730005e−005 w[402] =3.568554800150508e−005 w[403] = 2.443161189273522e−005 w[404] =1.334090914042349e−005 w[405] = 2.853437194757816e−006 w[406] =−1.039263591111469e−004 w[407] = 5.144969377044875e−005 w[408] =9.711681816385056e−005 w[409] = 2.472023910553232e−005 w[410] =5.397064424090302e−005 w[411] = 6.487880719449901e−005 w[412] =−5.192444140699947e−005 w[413] = −9.204876089551197e−005 w[414] =−1.815837353167847e−004 w[415] = −3.595054179561440e−004 w[416] =−5.901617707607606e−007 w[417] = 1.831121301698088e−004 w[418] =9.755685190624611e−005 w[419] = 6.606461762989423e−005 w[420] =3.799971890923797e−005 w[421] = 4.150075391929448e−005 w[422] =5.021905476506264e−005 w[423] = 5.861800137434713e−005 w[424] =2.126364641291926e−005 w[425] = 1.181077582797280e−004 w[426] =9.990757789944374e−005 w[427] = 1.035782617124906e−004 w[428] =8.870181845310037e−005 w[429] = 5.533953373249822e−005 w[430] =1.580188994455254e−005 w[431] = 1.277184430250593e−006 w[432] =5.009913312943629e−006 w[433] = 1.499170392246774e−005 w[434] =2:241545750231630e−005 w[435] = 3.628511258723260e−005 w[436] =2.406516798531014e−005 w[437] = 2.515118233957011e−005 w[438] =3.759629789955498e−005 w[439] = 5.408154543124121e−005 w[440] =4.493916063285122e−005 w[441] = 2.806963579578946e−005 w[442] =2.364518513682831e−005 w[443] = 1.260639764582286e−005 w[444] =−2.599467772603631e−008 w[445] = −1.774108392496017e−005 w[446] =−5.889276659458115e−006 w[447] = −4.663777919108619e−005 w[448] =−2.078886359425321e−004 w[449] = −2.131405580107761e−004 w[450] =−1.784192600231068e−004 w[451] = −1.744841754193053e−004 w[452] =−1.728672507238372e−004 w[453] = −1.885286127508226e−004 w[454] =−2.078299015661617e−004 w[455] = −2.123671573189573e−004 w[456] =−2.415166002501312e−004 w[457] = −2.217025456251449e−004 w[458] =−9.907630821710970e−005 w[459] = −8.039231481768845e−005 w[460] =−7.934509417722400e−005 w[461] = −5.874199358780108e−005 w[462] =−5.449816072329412e−005 w[463] = −4.489491034408147e−005 w[464] =−3.498285982359981e−005 w[465] = −1.748284921486958e−005 w[466] =−9.075430772832575e−006 w[467] = −1.052707430241351e−005 w[468] =−6.538878366985722e−006 w[469] = 2.206341308073472e−005 w[470] =1.769261935287328e−004 w[471] = 6.418658561385058e−005 w[472]−−8.882305312548962e−005 w[473] = −1.721347222211949e−005 w[474] =−6.093372716385583e−005 w[475] = −7.679955330373515e−005 w[476] =7.194151087015007e−005 w[477] = 7.245095937243279e−005 w[478] =7.870354371072524e−005 w[479] = 5.822201682995846e−004 w[480] =2.666444630171025e−004 w[481] = 7.872592352725688e−005 w[482] =7.095886893185526e−005 w[483] = 5.643103068471008e−005 w[484] =6.904415362098980e−005 w[485] = 4.694251739991356e−005 w[486] =3.367998338617662e−005 w[487] = 6.481921021601837e−005 w[488] =6.582328030188790e−005 w[489] = −4.256442530773449e−005 w[490] =4.939392400898679e−005 w[491] = 5.272982009116034e−005 w[492] =4.005269212731273e−005 w[493] = 2.461876679726978e−005 w[494] =4.469729032194765e−006 w[495] = 3.798519731621893e−007 w[496] =1.374896222030490e−006 w[497] = 3.965363805500215e−006 w[498] =7.300588863934780e−006 w[499] = 1.168894474770061e−005 w[500] =8.563819899447630e−006 w[501] = 8.975977837330335e−006 w[502] =2.800455533708622e−005 w[503] = 2.015445311139832e−005 w[504] =1.125134651175812e−005 w[505] = 5.869707265615299e−006 w[506] =1.013259758329981e−005 w[507] = 1.088325131492173e−005 w[508] =7.167101260771279e−006 w[509] = 4.840577540089826e−006 w[510] =−1.469933448634890e−005 w[511] = −8.010079089953001e−006 w[512] =−3.299004046633323e−005 w[513] = −4.373302115187172e−005 w[514] =−3.177468256997963e−005 w[515] = −2.976824036182567e−005 w[516] =−2.464228015326852e−005 w[517] = −1.606050838620834e−005 w[518] =−6.261944255489322e−006 w[519] = 4.591009581217994e−007 w[520] =1.395220723090848e−005 w[521] = 1.622786214398703e−005 w[522] =−2.043464113212971e−006 w[523] = −1.653463907257247e−006 w[524] =−1.551250801467300e−008 w[525] = −1.907927361317977e−006 w[526] =−9.607068622268791e−007 w[527] = −4.636105364510011e−007 w[528] =−2.765649762593200e−007 w[529] = −1.922074581855119e−006 w[530] =−9.897194091136331e−007 w[531] = −7.873304717454037e−008 w[532] =2.945239208477290e−008 w[533] = −2.757610624807679e−006 w[534] =−1.402925247695813e−005 w[535] = −9.388962780643742e−006 w[536] =2.068297421740023e−005 w[537] = 1.496435902895210e−007 w[538] =6.757014945674924e−009 w[539] = −2.778618354859861e−007 w[540] =−1.569003268449803e−006 w[541] = −1.089500601234349e−006 w[542] =−9.870547653835426e−007 w[543] = 3.867483283567218e−005 w[544] =−1.232693496472088e−005 w[545] = 9.464782951082177e−007 w[546] =8.254429452094225e−007 w[547] = 4.883304950437536e−007 w[548] =−2.066961713890010e−007 w[549] = 5.158212471036245e−009 w[550] =2.267731106642486e−007 w[551] = −4.880844550713951e−008 w[552] =3.361682183852576e−006 w[553] = 4.677015459111491e−006 w[554] =2.820292122791583e−008 w[555] = 5.143614846654519e−007 w[556] =3.818588614859347e−009 w[557] = 1.737276553950212e−007 w[558] =1.876022048145804e−007 w[559] = −2.986488593070417e−009 w[560] =−1.409927495646886e−008 w[561] = −6.977078748707401e−008 w[562] =−1.280675520205100e−008 w[563] = −2.222072007942510e−009 w[564] =−1.775191290895584e−009 w[565] = −1.686136654621906e−009 w[566] =5.818594642226675e−006 w[567] = 2.150883991167946e−006 w[568] =2.714879009950152e−007 w[569] = −2.567964804401197e−008 w[570] =2.041128570435378e−006 w[571] = 3.262753594084781e−006 w[572] =3.567581483749161e−006 w[573] = 4.083718802566134e−006 w[574] =5.364807253588177e−006 w[575] = 4.178050149840223e−006 w[576] =5.189086332701670e−006 w[577] = 3.357218747491756e−006 w[578] =6.310207878018869e−006 w[579] = 5.924001540927652e−006 w[580] =5.161606640348293e−006 w[581] = 3.377814811745950e−006 w[582] =1.323267689777069e−006 w[583] = −1.074716688428712e−007 w[584] =−3.561585382456484e−006 w[585] = −4.518603099564185e−006 w[586] =7.301956971603966e−007 w[587] = 5.891904775161025e−007 w[588] =2.801882088134371e−008 w[589] = 6.322770332405526e−007 w[590] =2.542598385847351e−007 w[591] = 1.272704908592385e−007 w[592] =8.226599990523664e−008 w[593] = 5.433718768789140e−007 w[594] =4.211177232106135e−007 w[595] = 3.552991527555180e−008 w[596] =−1.398913109540774e−008 w[597] = 1.356727552196146e−006 w[598] =−1.706941020342299e−005 w[599] = 1.013575160981381e−005 w[600] =−2.285562946018590e−005 w[601] = −8.908041185396514e−008 w[602] =−9.597515277415496e−009 w[603] = −3.225913527455964e−007 w[604] =1.070242712585309e−006 w[605] = 6.293002327021578e−007 w[606] =3.575650976036433e−007 w[607] = 2.722295965060517e−005 w[608] =8.676848186676888e−006 w[609] = 3.428660858940255e−007 w[610] =4.767793949944890e−007 w[611] = 3.330981930777764e−007 w[612] =2.399696144635756e−007 w[613] = 7.326611439066549e−009 w[614] =1.349943693297681e−007 w[615] = −5.393555749348494e−008 w[616] =3.629067065524143e−006 w[617] = −5.690530948134642e−006 w[618] =1.387566465624550e−008 w[619] = 2.40085172403935e−007 w[620] =1.723217058490933e−009 w[621] = 7.391973323448250e−008 w[622] =5.303527922331415e−008 w[623] = −8.883499047404846e−010 w[624] =−3.870536804891648e−009 w[625] = −1.846547564287500e−008 w[626] =−4.244090917065736e−009 w[627] = −4.013524925634108e−009 w[628] =−6.325664562585882e−010 w[629] = −6.025110605409611e−010 w[630] =1.620171502086309e−006 w[631] = 5.490569954646963e−007 w[632] =6.355303179925355e−008 w[633] = −5.426597100684762e−009 w[634] =4.292861814894369e−007 w[635] = 6.834209542421138e−007 w[636] =7.099633014995863e−007 w[637] = 8.109951846981774e−007 w[638] =4.118359768898598e−007 w[639] = 6.571760029213382e−007

Annex 2 |w[0]| = 1.129580193872797e−002 |w[1]| = 2.353059744904218e−002|w[2]| = 3.450718748721251e−002 |w[3]| = 4.634695977000525e−002 |w[4]| =5.918677345174197e−002 |w[5]| = 7.325978412117062e−002 |w[6]| =8.829745229234007e−002 |w[7]| = 1.042033024802571e−001 |w[8]| =1.206924277410051e−001 |w[9]| = 1.376149808913910e−001 |w[10]| =1.547461142258783e−001 |w[11]| = 1.719726384566089e−001 |w[12]| =1.891590407342011e−001 |w[13]| = 2.062605107774960e−001 |w[14]| =2.232276864673650e−001 |w[15]| = 2.400768261284114e−001 |w[16]| =2.568176309566753e−001 |w[17]| = 2.734977190313227e−001 |w[18]| =2.901491317310591e−001 |w[19]| = 3.068186515423912e−001 |w[20]| =3.235298682841570e−001 |w[21]| = 3.403074146062977e−001 |w[22]| =3.571527896130669e−001 |w[23]| = 3.740643974275026e−001 |w[24]| =3.910243970160607e−001 |w[25]| = 4.080154903861317e−001 |w[26]| =4.250144186334534e−001 |w[27]| = 4.420013942269341e−001 |w[28]| =4.589582896478246e−001 |w[29]| = 4.758753745532750e−001 |w[30]| =4.927463828072591e−001 |w[31]| = 5.095720854151864e−001 |w[32]| =5.263554446856779e−001 |w[33]| = 5.430990601899994e−001 |w[34]| =5.598052330684253e−001 |w[35]| = 5.764734796907189e−001 |w[36]| =5.930981800982896e−001 |w[37]| = 6.096690552916387e−001 |w[38]| =6.261725236758639e−001 |w[39]| = 6.425939632009995e−001 |w[40]| =6.589148753746076e−001 |w[41]| = 6.751199626157149e−001 |w[42]| =6.911981575264606e−001 |w[43]| = 7.071447728928043e−001 |w[44]| =7.229599104052475e−001 |w[45]| = 7.386515025302785e−001 |w[46]| =7.542294504292890e−001 |w[47]| = 7.697093346240386e−001 |w[48]| =7.851012620144958e−001 |w[49]| = 8.004165237845137e−001 |w[50]| =8.156523162880560e−001 |w[51]| = 8.308039608112368e−001 |w[52]| =8.458450064727010e−001 |w[53]| = 8.607492455327098e−001 |w[54]| =8.754640719350776e−001 |w[55]| = 8.899474405744183e−001 |w[56]| =9.041286138017367e−001 |w[57]| = 9.179666107725365e−001 |w[58]| =9.313874086278087e−001 |w[59]| = 9.443802853939540e−001 |w[60]| =9.568885413848645e−001 |w[61]| = 9.690016637782843e−001 |w[62]| =9.807691702375303e−001 |w[63]| = 9.927543720639498e−001 |w[64]| =1.001463112557766e+000 |w[65]| = 1.006893331637123e+000 |w[66]| =1.012508393574432e+000 |w[67]| = 1.017729040219375e+000 |w[68]| =1.022470190536100e+000 |w[69]| = 1.026615653698808e+000 |w[70]| =1.030198648769593e+000 |w[71]| = 1.033205850580933e+000 |w[72]| =1.035694432087486e+000 |w[73]| = 1.037683165297586e+000 |w[74]| =1.039227995800217e+000 |w[75]| = 1.040349586463588e+000 |w[76]| =1.041086497214721e+000 |w[77]| = 1.041443375950143e+000 |w[78]| =1.041434355650865e+000 |w[79]| = 1.041043184216171e+000 |w[80]| =1.040262316588456e+000 |w[81]| = 1.039061496136853e+000 |w[82]| =1.037422300157921e+000 |w[83]| = 1.035311720204252e+000 |w[84]| =1.032712952177121e+000 |w[85]| = 1.029600494883906e+000 |w[86]| =1.025966756910904e+000 |w[87]| = 1.021798805583990e+000 |w[88]| =1.017100128250049e+000 |w[89]| = 1.011867706519706e+000 |w[90]| =1.006109248754940e+000 |w[91]| = 9.998285752401580e−001 |w[92]| =9.930379854679836e−001 |w[93]| = 9.857387823493258e−001 |w[94]| =9.779405164766706e−001 |w[95]| = 9.696426101291272e−001 |w[96]| =9.608519516143015e−001 |w[97]| = 9.515674613550604e−001 |w[98]| =9.417975696327747e−001 |w[99]| = 9.315442093447622e−001 |w[100]| =9.208194746232827e−001 |w[101]| = 9.096310803629866e−001 |w[102]| =8.979959173503500e−001 |w[103]| = 8.859232320517536e−001 |w[104]| =8.734366852542127e−001 |w[105]| = 8.605542791988831e−001 |w[106]| =8.472987145504696e−001 |w[107]| = 8.336863467961255e−001 |w[108]| =8.197387292306723e−001 |w[109]| = 8.054701312929008e−001 |w[110]| =7.908995350037713e−001 |w[111]| = 7.760385598209244e−001 |w[112]| =7.609051036128973e−001 |w[113]| = 7.455111681431031e−001 |w[114]| =7.298745530879272e−001 |w[115]| = 7.140087729493950e−001 |w[116]| =6.979336851549095e−001 |w[117]| = 6.816667882498023e−001 |w[118]| =6.652304141388827e−001 |w[119]| = 6.486437667370537e−001 |w[120]| =6.319284031798550e−001 |w[121]| = 6.151031151692835e−001 |w[122]| =5.981877665956570e−001 |w[123]| = 5.811992722116214e−001 |w[124]| =5.641522833259215e−001 |w[125]| = 5.470652177576862e−001 |w[126]| =5.299509559653194e−001 |w[127]| = 5.128557121424191e−001 |w[128]| =4.956175421414453e−001 |w[129]| = 4.782650346610896e−001 |w[130]| =4.609828932783459e−001 |w[131]| = 4.437530233023859e−001 |w[132]| =4.265950246465440e−001 |w[133]| = 4.095160467543179e−001 |w[134]| =3.925409172155113e−001 |w[135]| = 3.756821671788237e−001 |w[136]| =3.589626517817934e−001 |w[137]| = 3.423942311297658e−001 |w[138]| =3.259993851088293e−001 |w[139]| = 3.097861805973821e−001 |w[140]| =2.937724988593393e−001 |w[141]| = 2.779637821990255e−001 |w[142]| =2.623749159488041e−001 |w[143]| = 2.470098299603623e−001 |w[144]| =2.318815478758375e−001 |w[145]| = 2.169925682529340e−001 |w[146]| =2.023548005388463e−001 |w[147]| = 1.879711746686855e−001 |w[148]| =1.738542127021508e−001 |w[149]| = 1.600061812296078e−001 |w[150]| =1.464389150679625e−001 |w[151]| = 1.331544923127771e−001 |w[152]| =1.201628679722633e−001 |w[153]| = 1.074630704470568e−001 |w[154]| =9.506966959632511e−002 |w[155]| = 8.298103104739203e−002 |w[156]| =7.120356992726613e−002 |w[157]| = 5.973741829536090e−002 |w[158]| =4.859005767016811e−002 |w[159]| = 3.775928110298274e−002 |w[160]| =2.726484300186575e−002 |w[161]| = 1.711323992709580e−002 |w[162]| =7.298197371320593e−003 |w[163]| = 2.184256929356781e−003 |w[164]| =1.132324047372148e−002 |w[165]| = 2.012236990754980e−002 |w[166]| =2.857528272530154e−002 |w[167]| = 3.666942822678171e−002 |w[168]| =4.439683978044157e−002 |w[169]| = 5.177964768870787e−002 |w[170]| =5.881296711410786e−002 |w[171]| = 6.550209046893848e−002 |w[172]| =7.184073822817207e−002 |w[173]| = 7.783299328224960e−002 |w[174]| =8.347150698567406e−002 |w[175]| = 8.875756217893037e−002 |w[176]| =9.368651761350569e−002 |w[177]| = 9.826251129465624e−002 |w[178]| =1.024804711677230e−001 |w[179]| = 1.063454554357498e−001 |w[180]| =1.098551252869576e−001 |w[181]| = 1.130180022553412e−001 |w[182]| =1.158358935177899e−001 |w[183]| = 1.183233335449968e−001 |w[184]| =1.204854506722672e−001 |w[185]| = 1.223371395264402e−001 |w[186]| =1.238868653862843e−001 |w[187]| = 1.251477258491527e−001 |w[188]| =1.261262023246478e−001 |w[189]| = 1.268280540744526e−001 |w[190]| =1.272498700590511e−001 |w[191]| = 1.273590703506806e−001 |w[192]| =1.274567595465545e−001 |w[193]| = 1.275561350483646e−001 |w[194]| =1.273648326872248e−001 |w[195]| = 1.269415772180714e−001 |w[196]| =1.262995646340671e−001 |w[197]| = 1.254605188749804e−001 |w[198]| =1.244269583009826e−001 |w[199]| = 1.232131583108813e−001 |w[200]| =1.218183974842866e−001 |w[201]| = 1.202545652840080e−001 |w[202]| =1.185243106889108e−001 |w[203]| = 1.166399102636992e−001 |w[204]| =1.146042249339280e−001 |w[205]| = 1.124296184976912e−001 |w[206]| =1.101215600923314e−001 |w[207]| = 1.076972053405737e−001 |w[208]| =1.051641975499523e−001 |w[209]| = 1.025397604985405e−001 |w[210]| =9.982957934346254e−002 |w[211]| = 9.705239536075722e−002 |w[212]| =9.421624116597689e−002 |w[213]| = 9.133590931873967e−002 |w[214]| =8.841813387276727e−002 |w[215]| = 8.547715661443602e−002 |w[216]| =8.251962055343706e−002 |w[217]| = 7.955570759229536e−002 |w[218]| =7,657649751612349e−002 |w[219]| = 7.360559211914287e−002 |w[220]| =7.064948295960993e−002 |w[221]| = 6.771675107480543e−002 |w[222]| =6.480448458935215e−002 |w[223]| = 6.192692754258131e−002 |w[224]| =5.911363249658311e−002 |w[225]| = 5.637219228757212e−002 |w[226]| =5.368313072045600e−002 |w[227]| = 5.105620793438655e−002 |w[228]| =4.849284995895640e−002 |w[229]| = 4.599068181839981e−002 |w[230]| =4.355568588898841e−002 |w[231]| = 4.125570251909672e−002 |w[232]| =3.907137550527191e−002 |w[233]| = 3.696342556744636e−002 |w[234]| =3.493300140502248e−002 |w[235]| = 3.298151059524886e−002 |w[236]| =3.110861245410919e−002 |w[237]| = 2.931525594774175e−002 |w[238]| =2.760090729801069e−002 |w[239]| = 2.597956638848436e−002 |w[240]| =2.443433592149451e−002 |w[241]| = 2.296470793543091e−002 |w[242]| =2.156304510969632e−002 |w[243]| = 2.023524610221679e−002 |w[244]| =1.897505817503749e−002 |w[245]| = 1.778248750467421e−002 |w[246]| =1.665187994388476e−002 |w[247]| = 1.557759513377242e−002 |w[248]| =1.456208586604537e−002 |w[249]| = 1.361072086117313e−002 |w[250]| =1.270747042064656e−002 |w[251]| = 1.186210743261470e−002 |w[252]| =1.106958962776399e−002 |w[253]| = 1.033126278863177e−002 |w[254]| =9.640298325700842e−003 |w[255]| = 8.996371481700806e−003 |w[256]| =8.407748878436545e−003 |w[257]| = 7.876393114319395e−003 |w[258]| =7.380543918629573e−003 |w[259]| = 6.925141135202262e−003 |w[260]| =6.500502521462604e−003 |w[261]| = 6.109178606718115e−003 |w[262]| =5.741103163221257e−003 |w[263]| = 5.394569608919965e−003 |w[264]| =5.063851046064050e−003 |w[265]| = 4.754191853611012e−003 |w[266]| =4.448993249380505e−003 |w[267]| = 4.133639756278191e−003 |w[268]| =3.811612348723333e−003 |w[269]| = 3.505531318950422e−003 |w[270]| =3.209092846617964e−003 |w[271]| = 2.927159436740159e−003 |w[272]| =2.653818578698405e−003 |w[273]| = 2.396404013961463e−003 |w[274]| =2.152379960589273e−003 |w[275]| = 1.924844672908215e−003 |w[276]| =1.699160580023900e−003 |w[277]| = 1.480542563288228e−003 |w[278]| =1.283280633901446e−003 |w[279]| = 1.131859661378862e−003 |w[280]| =9.730460256556873e−004 |w[281]| = 7.677634115875747e−004 |w[282]| =5.599347984905645e−004 |w[283]| = 3.337966579125254e−004 |w[284]| =9.099722643476421e−005 |w[285]| = 1.498231621816041e−004 |w[286]| =4.366447012116811e−004 |w[287]| = 6.307841647560053e−004 |w[288]| =6.150316826138937e−004 |w[289]| = 8.990255827053560e−004 |w[290]| =1.232134364570107e−003 |w[291]| = 1.471167206249042e−003 |w[292]| =1.697652664777771e−003 |w[293]| = 1.985825255428654e−003 |w[294]| =2.172866052963961e−003 |w[295]| = 1.812176023993582e−003 |w[296]| =1.344657262814793e−003 |w[297]| = 9.373975348172919e−004 |w[298]| =5.621720998949145e−004 |w[299]| = 2.048498552413189e−004 |w[300]| =2.004822830002534e−004 |w[301]| = 6.169854804735951e−004 |w[302]| =1.061498982103114e−003 |w[303]| = 1.594860949611097e−003 |w[304]| =2.124647831574725e−003 |w[305]| = 2.621537051750861e−003 |w[306]| =3.064311083207632e−003 |w[307]| = 3.460362845825662e−003 |w[308]| =3.794425324215804e−003 |w[309]| = 4.091032597247918e−003 |w[310]| =4.369553676668050e−003 |w[311]| = 4.554811297024067e−003 |w[312]| =4.663276675479689e−003 |w[313]| = 4.722567636185647e−003 |w[314]| =4.704321497976561e−003 |w[315]| = 4.636227793039124e−003 |w[316]| =4.517190210387324e−003 |w[317]| = 4.351667566540186e−003 |w[318]| =4.135130493071822e−003 |w[319]| = 3.870851645947402e−003 |w[320]| =3.597475533950260e−003 |w[321]| = 3.318857985461042e−003 |w[322]| =3.000422543655664e−003 |w[323]| = 2.658042081080524e−003 |w[324]| =2.292813563887493e−003 |w[325]| = 1.914114740669928e−003 |w[326]| =1.525818616748839e−003 |w[327]| = 1.1.56680209049319e−003 |w[328]| =7.804546272743493e−004 |w[329]| = 4.268574601396473e−004 |w[330]| =1.324291707264515e−004 |w[331]| = 1.218226450050751e−004 |w[332]| =3.189336138130849e−004 |w[333]| = 4.749931197951235e−004 |w[334]| =5.970696819774243e−004 |w[335]| = 6.673250213055329e−004 |w[336]| =6.887783835812338e−004 |w[337]| = 6.766320515830324e−004 |w[338]| =6.944123176012471e−004 |w[339]| = 7.139919634325070e−004 |w[340]| =7.154123487609100e−004 |w[341]| = 7.376101027486600e−004 |w[342]| =6.976561203768226e−004 |w[343]| = 5.721223454434728e−004 |w[344]| =2.934875643581191e−004 |w[345]| = 1.092526149391273e−004 |w[346]| =6.415402443848103e−004 |w[347]| = 1.194730618383423e−003 |w[348]| =1.557112059887280e−003 |w[349]| = 1.891971801393744e−003 |w[350]| =2.225524159129023e−003 |w[351]| = 2.530906981099261e−003 |w[352]| =2.719749515067397e−003 |w[353]| = 2.729136737522100e−003 |w[354]| =2.703019498899013e−003 |w[355]| = 2.630471852319136e−003 |w[356]| =2.470456304276468e−003 |w[357]| = 2.239142906871446e−003 |w[358]| =2.033465291493264e−003 |w[359]| = 1.948069005335563e−003 |w[360]| =1.725029670030533e−003 |w[361]| = 1.417366709895927e−003 |w[362]| =1.127141815310061e−003 |w[363]| = 8.089811988213151e−004 |w[364]| =4.708009521678285e−004 |w[365]| = 7.882620739833088e−005 |w[366]| =2.998739993995956e−004 |w[367]| = 4.733148292475610e−004 |w[368]| =5.791145447913150e−004 |w[369]| = 6.754935404082003e−004 |w[370]| =8.029620210721900e−004 |w[371]| = 9.726698841994444e−004 |w[372]| =1.196637962311630e−003 |w[373]| = 1.292865844760059e−003 |w[374]| =1.146268465739874e−003 |w[375]| = 1.040598055074471e−003 |w[376]| =9.767709065548874e−004 |w[377]| = 9.294665200453614e−004 |w[378]| =9.862027119530482e−004 |w[379]| = 1.047654674829846e−003 |w[380]| =1.099000599887377e−003 |w[381]| = 1.151795860160292e−003 |w[382]| =1.194743370333155e−003 |w[383]| = 1.250742797799558e−003 |w[384]| =1.287819050086379e−003 |w[385]| = 1.263569296641556e−003 |w[386]| =1.226113111394085e−003 |w[387]| = 1.177515087338257e−003 |w[388]| =1.122503050159859e−003 |w[389]| = 1.089428846944533e−003 |w[390]| =1.054963366189962e−003 |w[391]| = 9.019128558297515e−004 |w[392]| =7.847839620863715e−004 |w[393]| = 6.205675927856794e−004 |w[394]| =3.157663628445906e−004 |w[395]| = 2.556449844935384e−004 |w[396]| =2.520606580606257e−004 |w[397]| = 2.346980949474655e−004 |w[398]| =2.060394037017961e−004 |w[399]| = 1.635905995590986e−004 |w[400]| =1.176237128375623e−004 |w[401]| = 6.193369904730005e−005 |w[402]| =3.568554800150508e−005 |w[403]| = 2.443161189273522e−005 |w[404]| =1.334090914042349e−005 |w[405]| = 2.853437194757816e−006 |w[406]| =1.039263591111469e−004 |w[407]| = 5.144969377044875e−005 |w[408]| =9.711681816385056e−005 |w[409]| = 2.472023910553232e−005 |w[410]| =5.397064424090302e−005 |w[411]| = 6.487880719449901e−005 |w[412]| =5.192444140699947e−005 |w[413]| = 9.204876089551197e−005 |w[414]| =1.815837353167847e−004 |w[415]| = 3.595054179561440e−004 |w[416]| =5.901617707607606e−007 |w[417]| = 1.831121301698088e−004 |w[418]| =9.755685190624611e−005 |w[419]| = 6.606461762989423e−005 |w[420]| =3.799971890923797e−005 |w[421]| = 4.150075391929448e−005 |w[422]| =5.021905476506264e−005 |w[423]| = 5.861800137434713e−005 |w[424]| =2.126364641291926e−005 |w[425]| = 1.181077582797280e−004 |w[426]| =9.990757789944374e−005 |w[427]| = 1.035782617124906e−004 |w[428]| =8.870181845310037e−005 |w[429]| = 5.533953373249822e−005 |w[430]| =1.580188994455254e−005 |w[431]| = 1.277184430250593e−006 |w[432]| =5.009913312943629e−006 |w[433]| = 1.499170392246774e−005 |w[434]| =2.241545750231630e−005 |w[435]| = 3.628511258723260e−005 |w[436]| =2.406516798531014e−005 |w[437]| = 2.515118233957011e−005 |w[438]| =3.759629789955498e−005 |w[439]| = 5.408154543124121e−005 |w[440]| =4.493916063285122e−005 |w[441]| = 2.806963579578946e−005 |w[442]| =2.364518513682831e−005 |w[443]| = 1.260639764582286e−005 |w[444]| =2.599467772603631e−008 |w[445]| = 1.774108392496017e−005 |w[446]| =5.889276659458115e−006 |w[447]| = 4.663777919108619e−005 |w[448]| =2.078886359425321e−004 |w[449]| = 2.131405580107761e−004 |w[450]| =1.784192600231068e−004 |w[451]| = 1.744841754193053e−004 |w[452]| =1.728672507238372e−004 |w[453]| = 1.885286127508226e−004 |w[454]| =2.078299015661617e−004 |w[455]| = 2.123671573189573e−004 |w[456]| =2.415166002501312e−004 |w[457]| = 2.217025456251449e−004 |w[458]| =9.907630821710970e−005 |w[459]| = 8.039231481768845e−005 |w[460]| =7.934509417722400e−005 |w[461]| = 5.874199358780108e−005 |w[462]| =5.449816072329412e−005 |w[463]| = 4.489491034408147e−005 |w[464]| =3.498285982359981e−005 |w[465]| = 1.748284921486958e−005 |w[466]| =9.075430772832575e−006 |w[467]| = 1.052707430241351e−005 |w[468]| =6.538878366985722e−006 |w[469]| = 2.206341308073472e−005 |w[470]| =1.769261935287328e−004 |w[471]| = 6.418658561385058e−005 |w[472]| =8.882305312548962e−005 |w[473]| = 1.721347222211949e−005 |w[474]| =6.093372716385583e−005 |w[475]| = 7.679955330373515e−005 |w[476]| =7.194151087015007e−005 |w[477]| = 7.245095937243279e−005 |w[478]| =7.870354371072524e−005 |w[479]| = 5.822201682995846e−004 |w[480]| =2.666444630171025e−004 |w[481]| = 7.872592352725688e−005 |w[482]| =7.095886893185526e−005 |w[483]| = 5.643103068471008e−005 |w[484]| =6.904415362098980e−005 |w[485]| = 4.694251739991356e−005 |w[486]| =3.367998338617662e−005 |w[487]| = 6.481921021601837e−005 |w[488]| =6.582328030188790e−005 |w[489]| = 4.256442530773449e−005 |w[490]| =4.939392400898679e−005 |w[491]| = 5.272982009116034e−005 |w[492]| =4.005269212731273e−005 |w[493]| = 2.461876679726978e−005 |w[494]| =4.469729032194765e−006 |w[495]| = 3.798519731621893e−007 |w[496]| =1.374896222030490e−006 |w[497]| = 3.965363805500215e−006 |w[498]| =7.300588863934780e−006 |w[499]| = 1.168894474770061e−005 |w[500]| =8.563819899447630e−006 |w[501]| = 8.975977837330335e−006 |w[502]| =2.800455533708622e−005 |w[503]| = 2.015445311139832e−005 |w[504]| =1.125134651175812e−005 |w[505]| = 5.869707265615299e−006 |w[506]| =1.013259758329981e−005 |w[507]| = 1.088325131492173e−005 |w[508]| =7.167101260771279e−006 |w[509]| = 4.840577540089826e−006 |w[510]| =1.469933448634890e−005 |w[511]| = 8.010079089953001e−006 |w[512]| =3.299004046633323e−005 |w[513]| = 4.373302115187172e−005 |w[514]| =3.177468256997963e−005 |w[515]| = 2.976824036182567e−005 |w[516]| =2.464228015326852e−005 |w[517]| = 1.606050838620834e−005 |w[518]| =6.261944255489322e−006 |w[519]| = 4.591009581217994e−007 |w[520]| =1.395220723090848e−005 |w[521]| = 1.622786214398703e−005 |w[522]| =2.043464113212971e−006 |w[523]| = 1.653463907257247e−006 |w[524]| =1.551250801467300e−008 |w[525]| = 1.907927361317977e−006 |w[526]| =9.607068622268791e−007 |w[527]| = 4.636105364510011e−007 |w[528]| =2.765649762593200e−007 |w[529]| = 1.922074581855119e−006 |w[530]| =9.897194091136331e−007 |w[531]| = 7.873304717454037e−008 |w[532]| =2.945239208477290e−008 |w[533]| = 2.757610624807679e−006 |w[534]| =1.402925247695813e−005 |w[535]| = 9.388962780643742e−006 |w[536]| =2.068297421740023e−005 |w[537]| = 1.496435902895210e−007 |w[538]| =6.757014945674924e−009 |w[539]| = 2.778618354859861e−007 |w[540]| =1.569003268449803e−006 |w[541]| = 1.089500601234349e−006 |w[542]| =9.870547653835426e−007 |w[543]| = 3.867483283567218e−005 |w[544]| =1.232693496472088e−005 |w[545]| = 9.464782951082177e−007 |w[546]| =8.254429452094225e−007 |w[547]| = 4.883304950437536e−007 |w[548]| =2.066961713890010e−007 |w[549]| = 5.158212471036245e−009 |w[550]| =2.267731106642486e−007 |w[551]| = 4.880844550713951e−008 |w[552]| =3.361682183852576e−006 |w[553]| = 4.677015459111491e−006 |w[554]| =2.820292122791583e−008 |w[555]| = 5.143614846654519e−007 |w[556]| =3.818588614859347e−009 |w[557]| = 1.737276553950212e−007 |w[558]| =1.876022048145804e−007 |w[559]| = 2.986488593070417e−009 |w[560]| =1.409927495646886e−008 |w[561]| = 6.977078748707401e−008 |w[562]| =1.280675520205100e−008 |w[563]| = 2.222072007942510e−009 |w[564]| =1.775191290895584e−009 |w[565]| = 1.686136654621906e−009 |w[566]| =5.818594642226675e−006 |w[567]| = 2.150883991167946e−006 |w[568]| =2.714879009950152e−007 |w[569]| = 2.567964804401197e−008 |w[570]| =2.041128570435378e−006 |w[571]| = 3.262753594084781e−006 |w[572]| =3.567581483749161e−006 |w[573]| = 4.083718802566134e−006 |w[574]| =5.364807253588177e−006 |w[575]| = 4.178050149840223e−006 |w[576]| =5.189086332701670e−006 |w[577]| = 3.357218747491756e−006 |w[578]| =6.310207878018869e−006 |w[579]| = 5.924001540927652e−006 |w[580]| =5.161606640348293e−006 |w[581]| = 3.377814811745950e−006 |w[582]| =1.323267689777069e−006 |w[583]| = 1.074716688428712e−007 |w[584]| =3.561585382456484e−006 |w[585]| = 4.518603099564185e−006 |w[586]| =7.301956971603966e−007 |w[587]| = 5.891904775161025e−007 |w[588]| =2.801882088134371e−008 |w[589]| = 6.322770332405526e−007 |w[590]| =2.542598385847351e−007 |w[591]| = 1.272704908592385e−007 |w[592]| =8.226599990523664e−008 |w[593]| = 5.433718768789140e−007 |w[594]| =4.211177232106135e−007 |w[595]| = 3.552991527555180e−008 |w[596]| =1.398913109540774e−008 |w[597]| = 1.356727552196146e−006 |w[598]| =1.706941020342299e−005 |w[599]| = 1.013575160981381e−005 |w[600]| =2.285562946018590e−005 |w[601]| = 8.908041185396514e−008 |w[602]| =9.597515277415496e−009 |w[603]| = 3.225913527455964e−007 |w[604]| =1.070242712585309e−006 |w[605]| = 6.293002327021578e−007 |w[606]| =3.575650976036433e−007 |w[607]| = 2.722295965060517e−005 |w[608]| =8.676848186676888e−006 |w[609]| = 3.428660858940255e−007 |w[610]| =4.767793949944890e−007 |w[611]| = 3.330981930777764e−007 |w[612]| =2.399696144635756e−007 |w[613]| = 7.326611439066549e−009 |w[614]| =1.349943693297681e−007 |w[615]| = 5.393555749348494e−008 |w[616]| =3.629067065524143e−006 |w[617]| = 5.690530948134642e−006 |w[618]| =1.387566465624550e−008 |w[619]| = 2.443085172403935e−007 |w[620]| =1.723217058490933e−009 |w[621]| = 7.391973323448250e−008 |w[622]| =5.303527922331415e−008 |w[623]| = 8.883499047404846e−010 |w[624]| =3.870536804891648e−009 |w[625]| = 1.846547564287500e−008 |w[626]| =4.244090917065736e−009 |w[627]| = 4.013524925634108e−009 |w[628]| =6.325664562585882e−010 |w[629]| = 6.025110605409611e−010 |w[630]| =1.620171502086309e−006 |w[631]| = 5.490569954646963e−007 |w[632]| =6.355303179925355e−008 |w[633]| = 5.426597100684762e−009 |w[634]| =4.292861814894369e−007 |w[635]| = 6.834209542421138e−007 |w[636]| =7.099633014995863e−007 |w[637]| = 8.109951846981774e−007 |w[638]| =4.118359768898598e−007 |w[639]| = 6.571760029213382e−007

Annex 3 0.010 < w[0] < 0.012 0.023 < w[1] < 0.025 0.034 < w[2] < 0.0360.045 < w[3] < 0.047 0.058 < w[4] < 0.060 0.072 < w[5] < 0.074 0.087 <w[6] < 0.089 0.103 < w[7] < 0.105 0.120 < w[8] < 0.122 0.137 < w[9] <0.139 0.154 < w[10] < 0.156 0.171 < w[11] < 0.173 0.188 < w[12] < 0.1900.205 < w[13] < 0.207 0.222 < w[14] < 0.224 0.239 < w[15] < 0.241 0.256< w[16] < 0.258 0.272 < w[17] < 0.274 0.289 < w[18] < 0.291 0.306 <w[19] < 0.308 0.323 < w[20] < 0.325 0.339 < w[21] < 0.341 0.356 < w[22]< 0.358 0.373 < w[23] < 0.375 0.390 < w[24] < 0.392 0.407 < w[25] <0.409 0.424 < w[26] < 0.426 0.441 < w[27] < 0.443 0.458 < w[28] < 0.4600.475 < w[29] < 0.477 0.492 < w[30] < 0.494 0.509 < w[31] < 0.511 0.525< w[32] < 0.527 0.542 < w[33] < 0.544 0.559 < w[34] < 0.561 0.575 <w[35] < 0.577 0.592 < w[36] < 0.594 0.609 < w[37] < 0.611 0.625 < w[38]< 0.627 0.642 < w[39] < 0.644 0.658 < w[40] < 0.660 0.674 < w[41] <0.676 0.690 < w[42] < 0.692 0.706 < w[43] < 0.708 0.722 < w[44] < 0.7240.738 < w[45] < 0.740 0.753 < w[46] < 0.755 0.769 < w[47] < 0.771 0.784< w[48] < 0.786 0.799 < w[49] < 0.801 0.815 < w[50] < 0.817 0.830 <w[51] < 0.832 0.845 < w[52] < 0.847 0.860 < w[53] < 0.862 0.874 < w[54]< 0.876 0.889 < w[55] < 0.891 0.903 < w[56] < 0.905 0.917 < w[57] <0.919 0.930 < w[58] < 0.932 0.943 < w[59] < 0.945 0.956 < w[60] < 0.9580.968 < w[61] < 0.970 0.980 < w[62] < 0.982 0.992 < w[63] < 0.994 1.000< w[64] < 1.002 1.006 < w[65] < 1.008 1.012 < w[66] < 1.014 1.017 <w[67] < 1.019 1.021 < w[68] < 1.023 1.026 < w[69] < 1.028 1.029 < w[70]< 1.031 1.032 < w[71] < 1.034 1.035 < w[72] < 1.037 1.037 < w[73] <1.039 1.038 < w[74] < 1.040 1.039 < w[75] < 1.041 1.040 < w[76] < 1.0421.040 < w[77] < 1.042 1.040 < w[78] < 1.042 1.040 < w[79] < 1.042 1.039< w[80] < 1.041 1.038 < w[81] < 1.040 1.036 < w[82] < 1.038 1.034 <w[83] < 1.036 1.032 < w[84] < 1.034 1.029 < w[85] < 1.031 1.025 < w[86]< 1.027 1.021 < w[87] < 1.023 1.016 < w[88] < 1.018 1.011 < w[89] <1.013 1.005 < w[90] < 1.007 0.999 < w[91] < 1.001 0.992 < w[92] < 0.9940.985 < w[93] < 0.987 0.977 < w[94] < 0.979 0.969 < w[95] < 0.971 0.960< w[96] < 0.962 0.951 < w[97] < 0.953 0.941 < w[98] < 0.943 0.931 <w[99] < 0.933 0.920 < w[100] < 0.922 0.909 < w[101] < 0.911 0.897 <w[102] < 0.899 0.885 < w[103] < 0.887 0.872 < w[104] < 0.874 0.860 <w[105] < 0.862 0.846 < w[106] < 0.848 0.833 < w[107] < 0.835 0.819 <w[108] < 0.821 0.804 < w[109] < 0.806 0.790 < w[110] < 0.792 0.775 <w[111] < 0.777 0.760 < w[112] < 0.762 0.745 < w[113] < 0.747 0.729 <w[114] < 0.731 0.713 < w[115] < 0.715 0.697 < w[116] < 0.699 0.681 <w[117] < 0.683 0.664 < w[118] < 0.666 0.648 < w[119] < 0.650 0.631 <w[120] < 0.633 0.614 < w[121] < 0.616 0.597 < w[122] < 0.599 0.580 <w[123] < 0.582 0.563 < w[124] < 0.565 0.546 < w[125] < 0.548 0.529 <w[126] < 0.531 0.512 < w[127] < 0.514 −0.497 < w[128] < −0.495 −0.479 <w[129] < −0.477 −0.462 < w[130] < −0.460 −0.445 < w[131] < −0.443 −0.428< w[132] < −0.426 −0.411 < w[133] < −0.409 −0.394 < w[134] < −0.392−0.377 < w[135] < −0.375 −0.360 < w[136] < −0.358 −0.343 < w[137] <−0.341 −0.327 < w[138] < −0.325 −0.311 < w[139] < −0.309 −0.295 < w[140]< −0.293 −0.279 < w[141] < −0.277 −0.263 < w[142] < −0.261 −0.248 <w[143] < −0.246 −0.233 < w[144] < −0.231 −0.218 < w[145] < −0.216 −0.203< w[146] < −0.201 −0.189 < w[147] < −0.187 −0.175 < w[148] < −0.173−0.161 < w[149] < −0.159 −0.147 < w[150] < −0.145 −0.134 < w[151] <−0.132 −0.121 < w[152] < −0.119 −0.108 < w[153] < −0.106 −0.096 < w[154]< −0.094 −0.084 < w[155] < −0.082 −0.072 < w[156] < −0.070 −0.061 <w[157] < −0.059 −0.050 < w[158] < −0.048 −0.039 < w[159] < −0.037 −0.028< w[160] < −0.026 −0.018 < w[161] < −0.016 −0.008 < w[162] < −0.0060.001 < w[163] < 0.003 0.010 < w[164] < 0.012 0.019 < w[165] < 0.0210.028 < w[166] < 0.030 0.036 < w[167] < 0.038 0.043 < w[168] < 0.0450.051 < w[169] < 0.053 0.058 < w[170] < 0.060 0.065 < w[171] < 0.0670.071 < w[172] < 0.073 0.077 < w[173] < 0.079 0.082 < w[174] < 0.0840.088 < w[175] < 0.090 0.093 < w[176] < 0.095 0.097 < w[177] < 0.0990.101 < w[178] < 0.103 0.105 < w[179] < 0.107 0.109 < w[180] < 0.1110.112 < w[181] < 0.114 0.115 < w[182] < 0.117 0.117 < w[183] < 0.1190.119 < w[184] < 0.121 0.121 < w[185] < 0.123 0.123 < w[186] < 0.1250.124 < w[187] < 0.126 0.125 < w[188] < 0.127 0.126 < w[189] < 0.1280.126 < w[190] < 0.128 0.126 < w[191] < 0.128 0.126 < w[192] < 0.1280.127 < w[193] < 0.129 0.126 < w[194] < 0.128 0.126 < w[195] < 0.1280.125 < w[196] < 0.127 0.124 < w[197] < 0.126 0.123 < w[198] < 0.1250.122 < w[199] < 0.124 0.121 < w[200] < 0.123 0.119 < w[201] < 0.1210.118 < w[202] < 0.120 0.116 < w[203] < 0.118 0.114 < w[204] < 0.1160.111 < w[205] < 0.113 0.109 < w[206] < 0.111 0.107 < w[207] < 0.1090.104 < w[208] < 0.106 0.102 < w[209] < 0.104 0.099 < w[210] < 0.1010.096 < w[211] < 0.098 0.093 < w[212] < 0.095 0.090 < w[213] < 0.0920.087 < w[214] < 0.089 0.084 < w[215] < 0.086 0.082 < w[216] < 0.0840.079 < w[217] < 0.081 0.076 < w[218] < 0.078 0.073 < w[219] < 0.0750.070 < w[220] < 0.072 0.067 < w[221] < 0.069 0.064 < w[222] < 0.0660.061 < w[223] < 0.063 0.058 < w[224] < 0.060 0.055 < w[225] < 0.0570.053 < w[226] < 0.055 0.050 < w[227] < 0.052 0.047 < w[228] < 0.0490.045 < w[229] < 0.047 0.043 < w[230] < 0.045 0.040 < w[231] < 0.0420.038 < w[232] < 0.040 0.036 < w[233] < 0.038 0.034 < w[234] < 0.0360.032 < w[235] < 0.034 0.030 < w[236] < 0.032 0.028 < w[237] < 0.0300.027 < w[238] < 0.029 0.025 < w[239] < 0.027 0.023 < w[240] < 0.0250.022 < w[241] < 0.024 0.021 < w[242] < 0.023 0.019 < w[243] < 0.0210.018 < w[244] < 0.020 0.017 < w[245] < 0.019 0.016 < w[246] < 0.0180.015 < w[247] < 0.017 0.014 < w[248] < 0.016 0.013 < w[249] < 0.0150.012 < w[250] < 0.014 0.011 < w[251] < 0.013 0.010 < w[252] < 0.0120.009 < w[253] < 0.011 0.009 < w[254] < 0.011 0.008 < w[255] < 0.010−0.009 < w[256] < −0.007 −0.009 < w[257] < −0.007 −0.008 < w[258] <−0.006 −0.008 < w[259] < −0.006 −0.008 < w[260] < −0.006 −0.007 < w[261]< −0.005 −0.007 < w[262] < −0.005 −0.006 < w[263] < −0.004 −0.006 <w[264] < −0.004 −0.006 < w[265] < −0.004 −0.005 < w[266] < −0.003 −0.005< w[267] < −0.003 −0.005 < w[268] < −0.003 −0.005 < w[269] < −0.003−0.004 < w[270] < −0.002 −0.004 < w[271] < −0.002 −0.004 < w[272] <−0.002 −0.003 < w[273] < −0.001 −0.003 < w[274] < −0.001 −0.003 < w[275]< −0.001 −0.003 < w[276] < −0.001 −0.002 < w[277] < 0.000 −0.002 <w[278] < 0.000 −0.002 < w[279] < 0.000 −0.002 < w[280] < 0.000 −0.002 <w[281] < 0.000 −0.002 < w[282] < 0.000 −0.001 < w[283] < 0.001 −0.001 <w[284] < 0.001 −0.001 < w[285] < 0.001 −0.001 < w[286] < 0.001 0.000 <w[287] < 0.002 0.000 < w[288] < 0.002 0.000 < w[289] < 0.002 0.000 <w[290] < 0.002 0.000 < w[291] < 0.002 0.001 < w[292] < 0.003 0.001 <w[293] < 0.003 0.001 < w[294] < 0.003 0.001 < w[295] < 0.003 0.000 <w[296] < 0.002 0.000 < w[297] < 0.002 0.000 < w[298] < 0.002 −0.001 <w[299] < 0.001 −0.001 < w[300] < 0.001 −0.002 < w[301] < 0.000 −0.002 <w[302] < 0.000 −0.003 < w[303] < −0.001 −0.003 < w[304] < −0.001 −0.004< w[305] < −0.002 −0.004 < w[306] < −0.002 −0.004 < w[307] < −0.002−0.005 < w[308] < −0.003 −0.005 < w[309] < −0.003 −0.005 < w[310] <−0.003 −0.006 < w[311] < −0.004 −0.006 < w[312] < −0.004 −0.006 < w[313]< −0.004 −0.006 < w[314] < −0.004 −0.006 < w[315] < −0.004 −0.006 <w[316] < −0.004 −0.005 < w[317] < −0.003 −0.005 < w[318] < −0.003 −0.005< w[319] < −0.003 −0.005 < w[320] < −0.003 −0.004 < w[321] < −0.002−0.004 < w[322] < −0.002 −0.004 < w[323] < −0.002 −0.003 < w[324] <−0.001 −0.003 < w[325] < −0.001 −0.003 < w[326] < −0.001 −0.002 < w[327]< 0.000 −0.002 < w[328] < 0.000 −0.001 < w[329] < 0.001 −0.001 < w[330]< 0.001 −0.001 < w[331] < 0.001 −0.001 < w[332] < 0.001 −0.001 < w[333]< 0.001 0.000 < w[334] < 0.002 0.000 < w[335] < 0.002 0.000 < w[336] <0.002 0.000 < w[337] < 0.002 0.000 < w[338] < 0.002 0.000 < w[339] <0.002 0.000 < w[340] < 0.002 0.000 < w[341] < 0.002 0.000 < w[342] <0.002 0.000 < w[343] < 0.002 −0.001 < w[344] < 0.001 −0.001 < w[345] <0.001 0.000 < w[346] < 0.002 0.000 < w[347] < 0.002 0.001 < w[348] <0.003 0.001 < w[349] < 0.003 0.001 < w[350] < 0.003 0.002 < w[351] <0.004 0.002 < w[352] < 0.004 0.002 < w[353] < 0.004 0.002 < w[354] <0.004 0.002 < w[355] < 0.004 0.001 < w[356] < 0.003 0.001 < w[357] <0.003 0.001 < w[358] < 0.003 0.001 < w[359] < 0.003 0.001 < w[360] <0.003 0.000 < w[361] < 0.002 0.000 < w[362] < 0.002 0.000 < w[363] <0.002 −0.001 < w[364] < 0.001 −0.001 < w[365] < 0.001 −0.001 < w[366] <0.001 −0.001 < w[367] < 0.001 −0.002 < w[368] < 0.000 −0.002 < w[369] <0.000 −0.002 < w[370] < 0.000 −0.002 < w[371] < 0.000 −0.002 < w[372] <0.000 −0.002 < w[373] < 0.000 −0.002 < w[374] < 0.000 −0.002 < w[375] <0.000 −0.002 < w[376] < 0.000 −0.002 < w[377] < 0.000 −0.002 < w[378] <0.000 −0.002 < w[379] < 0.000 −0.002 < w[380] < 0.000 −0.002 < w[381] <0.000 −0.002 < w[382] < 0.000 −0.002 < w[383] < 0.000 0.000 < w[384] <0.002 0.000 < w[385] < 0.002 0.000 < w[386] < 0.002 0.000 < w[387] <0.002 0.000 < w[388] < 0.002 0.000 < w[389] < 0.002 0.000 < w[390] <0.002 0.000 < w[391] < 0.002 0.000 < w[392] < 0.002 0.000 < w[393] <0.002 −0.001 < w[394] < 0.001 −0.001 < w[395] < 0.001 −0.001 < w[396] <0.001 −0.001 < w[397] < 0.001 −0.001 < w[398] < 0.001 −0.001 < w[399] <0.001 −0.001 < w[400] < 0.001 −0.001 < w[401] < 0.001 −0.001 < w[402] <0.001 −0.001 < w[403] < 0.001 −0.001 < w[404] < 0.001 −0.001 < w[405] <0.001 −0.001 < w[406] < 0.001 −0.001 < w[407] < 0.001 −0.001 < w[408] <0.001 −0.001 < w[409] < 0.001 −0.001 < w[410] < 0.001 −0.001 < w[411] <0.001 −0.001 < w[412] < 0.001 −0.001 < w[413] < 0.001 −0.001 < w[414] <0.001 −0.001 < w[415] < 0.001 −0.001 < w[416] < 0.001 −0.001 < w[417] <0.001 −0.001 < w[418] < 0.001 −0.001 < w[419] < 0.001 −0.001 < w[420] <0.001 −0.001 < w[421] < 0.001 −0.001 < w[422] < 0.001 −0.001 < w[423] <0.001 −0.001 < w[424] < 0.001 −0.001 < w[425] < 0.001 −0.001 < w[426] <0.001 −0.001 < w[427] < 0.001 −0.001 < w[428] < 0.001 −0.001 < w[429] <0.001 −0.001 < w[430] < 0.001 −0.001 < w[431] < 0.001 −0.001 < w[432] <0.001 −0.001 < w[433] < 0.001 −0.001 < w[434] < 0.001 −0.001 < w[435] <0.001 −0.001 < w[436] < 0.001 −0.001 < w[437] < 0.001 −0.001 < w[438] <0.001 −0.001 < w[439] < 0.001 −0.001 < w[440] < 0.001 −0.001 < w[441] <0.001 −0.001 < w[442] < 0.001 −0.001 < w[443] < 0.001 −0.001 < w[444] <0.001 −0.001 < w[445] < 0.001 −0.001 < w[446] < 0.001 −0.001 < w[447] <0.001 −0.001 < w[448] < 0.001 −0.001 < w[449] < 0.001 −0.001 < w[450] <0.001 −0.001 < w[451] < 0.001 −0.001 < w[452] < 0.001 −0.001 < w[453] <0.001 −0.001 < w[454] < 0.001 −0.001 < w[455] < 0.001 −0.001 < w[456] <0.001 −0.001 < w[457] < 0.001 −0.001 < w[458] < 0.001 −0.001 < w[459] <0.001 −0.001 < w[460] < 0.001 −0.001 < w[461] < 0.001 −0.001 < w[462] <0.001 −0.001 < w[463] < 0.001 −0.001 < w[464] < 0.001 −0.001 < w[465] <0.001 −0.001 < w[466] < 0.001 −0.001 < w[467] < 0.001 −0.001 < w[468] <0.001 −0.001 < w[469] < 0.001 −0.001 < w[470] < 0.001 −0.001 < w[471] <0.001 −0.001 < w[472] < 0.001 −0.001 < w[473] < 0.001 −0.001 < w[474] <0.001 −0.001 < w[475] < 0.001 −0.001 < w[476] < 0.001 −0.001 < w[477] <0.001 −0.001 < w[478] < 0.001 0.000 < w[479] < 0.002 −0.001 < w[480] <0.001 −0.001 < w[481] < 0.001 −0.001 < w[482] < 0.001 −0.001 < w[483] <0.001 −0.001 < w[484] < 0.001 −0.001 < w[485] < 0.001 −0.001 < w[486] <0.001 −0.001 < w[487] < 0.001 −0.001 < w[488] < 0.001 −0.001 < w[489] <0.001 −0.001 < w[490] < 0.001 −0.001 < w[491] < 0.001 −0.001 < w[492] <0.001 −0.001 < w[493] < 0.001 −0.001 < w[494] < 0.001 −0.001 < w[495] <0.001 −0.001 < w[496] < 0.001 −0.001 < w[497] < 0.001 −0.001 < w[498] <0.001 −0.001 < w[499] < 0.001 −0.001 < w[500] < 0.001 −0.001 < w[501] <0.001 −0.001 < w[502] < 0.001 −0.001 < w[503] < 0.001 −0.001 < w[504] <0.001 −0.001 < w[505] < 0.001 −0.001 < w[506] < 0.001 −0.001 < w[507] <0.001 −0.001 < w[508] < 0.001 −0.001 < w[509] < 0.001 −0.001 < w[510] <0.001 −0.001 < w[511] < 0.001 −0.001 < w[512] < 0.001 −0.001 < w[513] <0.001 −0.001 < w[514] < 0.001 −0.001 < w[515] < 0.001 −0.001 < w[516] <0.001 −0.001 < w[517] < 0.001 −0.001 < w[518] < 0.001 −0.001 < w[519] <0.001 −0.001 < w[520] < 0.001 −0.001 < w[521] < 0.001 −0.001 < w[522] <0.001 −0.001 < w[523] < 0.001 −0.001 < w[524] < 0.001 −0.001 < w[525] <0.001 −0.001 < w[526] < 0.001 −0.001 < w[527] < 0.001 −0.001 < w[528] <0.001 −0.001 < w[529] < 0.001 −0.001 < w[530] < 0.001 −0.001 < w[531] <0.001 −0.001 < w[532] < 0.001 −0.001 < w[533] < 0.001 −0.001 < w[534] <0.001 −0.001 < w[535] < 0.001 −0.001 < w[536] < 0.001 −0.001 < w[537] <0.001 −0.001 < w[538] < 0.001 −0.001 < w[539] < 0.001 −0.001 < w[540] <0.001 −0.001 < w[541] < 0.001 −0.001 < w[542] < 0.001 −0.001 < w[543] <0.001 −0.001 < w[544] < 0.001 −0.001 < w[545] < 0.001 −0.001 < w[546] <0.001 −0.001 < w[547] < 0.001 −0.001 < w[548] < 0.001 −0.001 < w[549] <0.001 −0.001 < w[550] < 0.001 −0.001 < w[551] < 0.001 −0.001 < w[552] <0.001 −0.001 < w[553] < 0.001 −0.001 < w[554] < 0.001 −0.001 < w[555] <0.001 −0.001 < w[556] < 0.001 −0.001 < w[557] < 0.001 −0.001 < w[558] <0.001 −0.001 < w[559] < 0.001 −0.001 < w[560] < 0.001 −0.001 < w[561] <0.001 −0.001 < w[562] < 0.001 −0.001 < w[563] < 0.001 −0.001 < w[564] <0.001 −0.001 < w[565] < 0.001 −0.001 < w[566] < 0.001 −0.001 < w[567] <0.001 −0.001 < w[568] < 0.001 −0.001 < w[569] < 0.001 −0.001 < w[570] <0.001 −0.001 < w[571] < 0.001 −0.001 < w[572] < 0.001 −0.001 < w[573] <0.001 −0.001 < w[574] < 0.001 −0.001 < w[575] < 0.001 −0.001 < w[576] <0.001 −0.001 < w[577] < 0.001 −0.001 < w[578] < 0.001 −0.001 < w[579] <0.001 −0.001 < w[580] < 0.001 −0.001 < w[581] < 0.001 −0.001 < w[582] <0.001 −0.001 < w[583] < 0.001 −0.001 < w[584] < 0.001 −0.001 < w[585] <0.001 −0.001 < w[586] < 0.001 −0.001 < w[587] < 0.001 −0.001 < w[588] <0.001 −0.001 < w[589] < 0.001 −0.001 < w[590] < 0.001 −0.001 < w[591] <0.001 −0.001 < w[592] < 0.001 −0.001 < w[593] < 0.001 −0.001 < w[594] <0.001 −0.001 < w[595] < 0.001 −0.001 < w[596] < 0.001 −0.001 < w[597] <0.001 −0.001 < w[598] < 0.001 −0.001 < w[599] < 0.001 −0.001 < w[600] <0.001 −0.001 < w[601] < 0.001 −0.001 < w[602] < 0.001 −0.001 < w[603] <0.001 −0.001 < w[604] < 0.001 −0.001 < w[605] < 0.001 −0.001 < w[606] <0.001 −0.001 < w[607] < 0.001 −0.001 < w[608] < 0.001 −0.001 < w[609] <0.001 −0.001 < w[610] < 0.001 −0.001 < w[611] < 0.001 −0.001 < w[612] <0.001 −0.001 < w[613] < 0.001 −0.001 < w[614] < 0.001 −0.001 < w[615] <0.001 −0.001 < w[616] < 0.001 −0.001 < w[617] < 0.001 −0.001 < w[618] <0.001 −0.001 < w[619] < 0.001 −0.001 < w[620] < 0.001 −0.001 < w[621] <0.001 −0.001 < w[622] < 0.001 −0.001 < w[623] < 0.001 −0.001 < w[624] <0.001 −0.001 < w[625] < 0.001 −0.001 < w[626] < 0.001 −0.001 < w[627] <0.001 −0.001 < w[628] < 0.001 −0.001 < w[629] < 0.001 −0.001 < w[630] <0.001 −0.001 < w[631] < 0.001 −0.001 < w[632] < 0.001 −0.001 < w[633] <0.001 −0.001 < w[634] < 0.001 −0.001 < w[635] < 0.001 −0.001 < w[636] <0.001 −0.001 < w[637] < 0.001 −0.001 < w[638] < 0.001 −0.001 < w[639] <0.001

Annex 4 0.010 < |w[0]| < 0.012 0.023 < |w[1]| < 0.025 0.034 < |w[2]| <0.036 0.045 < |w[3]| < 0.047 0.058 < |w[4]| < 0.060 0.072 < |w[5]| <0.074 0.087 < |w[6]| < 0.089 0.103 < |w[7]| < 0.105 0.120 < |w[8]| <0.122 0.137 < |w[9]| < 0.139 0.154 < |w[10]| < 0.156 0.171 < |w[11]| <0.173 0.188 < |w[12]| < 0.190 0.205 < |w[13]| < 0.207 0.222 < |w[14]| <0.224 0.239 < |w[15]| < 0.241 0.256 < |w[16]| < 0.258 0.272 < |w[17]| <0.274 0.289 < |w[18]| < 0.291 0.306 < |w[19]| < 0.308 0.323 < |w[20]| <0.325 0.339 < |w[21]| < 0.341 0.356 < |w[22]| < 0.358 0.373 < |w[23]| <0.375 0.390 < |w[24]| < 0.392 0.407 < |w[25]| < 0.409 0.424 < |w[26]| <0.426 0.441 < |w[27]| < 0.443 0.458 < |w[28]| < 0.460 0.475 < |w[29]| <0.477 0.492 < |w[30]| < 0.494 0.509 < |w[31]| < 0.511 0.525 < |w[32]| <0.527 0.542 < |w[33]| < 0.544 0.559 < |w[34]| < 0.561 0.575 < |w[35]| <0.577 0.592 < |w[36]| < 0.594 0.609 < |w[37]| < 0.611 0.625 < |w[38]| <0.627 0.642 < |w[39]| < 0.644 0.658 < |w[40]| < 0.660 0.674 < |w[41]| <0.676 0.690 < |w[42]| < 0.692 0.706 < |w[43]| < 0.708 0.722 < |w[44]| <0.724 0.738 < |w[45]| < 0.740 0.753 < |w[46]| < 0.755 0.769 < |w[47]| <0.771 0.784 < |w[48]| < 0.786 0.799 < |w[49]| < 0.801 0.815 < |w[50]| <0.817 0.830 < |w[51]| < 0.832 0.845 < |w[52]| < 0.847 0.860 < |w[53]| <0.862 0.874 < |w[54]| < 0.876 0.889 < |w[55]| < 0.891 0.903 < |w[56]| <0.905 0.917 < |w[57]| < 0.919 0.930 < |w[58]| < 0.932 0.943 < |w[59]| <0.945 0.956 < |w[60]| < 0.958 0.968 < |w[61]| < 0.970 0.980 < |w[62]| <0.982 0.992 < |w[63]| < 0.994 1.000 < |w[64]| < 1.002 1.006 < |w[65]| <1.008 1.012 < |w[66]| < 1.014 1.017 < |w[67]| < 1.019 1.021 < |w[68]| <1.023 1.026 < |w[69]| < 1.028 1.029 < |w[70]| < 1.031 1.032 < |w[71]| <1.034 1.035 < |w[72]| < 1.037 1.037 < |w[73]| < 1.039 1.038 < |w[74]| <1.040 1.039 < |w[75]| < 1.041 1.040 < |w[76]| < 1.042 1.040 < |w[77]| <1.042 1.040 < |w[78]| < 1.042 1.040 < |w[79]| < 1.042 1.039 < |w[80]| <1.041 1.038 < |w[81]| < 1.040 1.036 < |w[82]| < 1.038 1.034 < |w[83]| <1.036 1.032 < |w[84]| < 1.034 1.029 < |w[85]| < 1.031 1.025 < |w[86]| <1.027 1.021 < |w[87]| < 1.023 1.016 < |w[88]| < 1.018 1.011 < |w[89]| <1.013 1.005 < |w[90]| < 1.007 0.999 < |w[91]| < 1.001 0.992 < |w[92]| <0.994 0.985 < |w[93]| < 0.987 0.977 < |w[94]| < 0.979 0.969 < |w[95]| <0.971 0.960 < |w[96]| < 0.962 0.951 < |w[97]| < 0.953 0.941 < |w[98]| <0.943 0.931 < |w[99]| < 0.933 0.920 < |w[100]| < 0.922 0.909 < |w[101]|< 0.911 0.897 < |w[102]| < 0.899 0.885 < |w[103]| < 0.887 0.872 <|w[104]| < 0.874 0.860 < |w[105]| < 0.862 0.846 < |w[106]| < 0.848 0.833< |w[107]| < 0.835 0.819 < |w[108]| < 0.821 0.804 < |w[109]| < 0.8060.790 < |w[110]| < 0.792 0.775 < |w[111]| < 0.777 0.760 < |w[112]| <0.762 0.745 < |w[113]| < 0.747 0.729 < |w[114]| < 0.731 0.713 < |w[115]|< 0.715 0.697 < |w[116]| < 0.699 0.681 < |w[117]| < 0.683 0.664 <|w[118]| < 0.666 0.648 < |w[119]| < 0.650 0.631 < |w[120]| < 0.633 0.614< |w[121]| < 0.616 0.597 < |w[122]| < 0.599 0.580 < |w[123]| < 0.5820.563 < |w[124]| < 0.565 0.546 < |w[125]| < 0.548 0.529 < |w[126]| <0.531 0.512 < |w[127]| < 0.514 0.495 < |w[128]| < 0.497 0.477 < |w[129]|< 0.479 0.460 < |w[130]| < 0.462 0.443 < |w[131]| < 0.445 0.426 <|w[132]| < 0.428 0.409 < |w[133]| < 0.411 0.392 < |w[134]| < 0.394 0.375< |w[135]| < 0.377 0.358 < |w[136]| < 0.360 0.341 < |w[137]| < 0.3430.325 < |w[138]| < 0.327 0.309 < |w[139]| < 0.311 0.293 < |w[140]| <0.295 0.277 < |w[141]| < 0.279 0.261 < |w[142]| < 0.263 0.246 < |w[143]|< 0.248 0.231 < |w[144]| < 0.233 0.216 < |w[145]| < 0.218 0.201 <|w[146]| < 0.203 0.187 < |w[147]| < 0.189 0.173 < |w[148]| < 0.175 0.159< |w[149]| < 0.161 0.145 < |w[150]| < 0.147 0.132 < |w[151]| < 0.1340.119 < |w[152]| < 0.121 0.106 < |w[153]| < 0.108 0.094 < |w[154]| <0.096 0.082 < |w[155]| < 0.084 0.070 < |w[156]| < 0.072 0.059 < |w[157]|< 0.061 0.048 < |w[158]| < 0.050 0.037 < |w[159]| < 0.039 0.026 <|w[160]| < 0.028 0.016 < |w[161]| < 0.018 0.006 < |w[162]| < 0.008 0.001< |w[163]| < 0.003 0.010 < |w[164]| < 0.012 0.019 < |w[165]| < 0.0210.028 < |w[166]| < 0.030 0.036 < |w[167]| < 0.038 0.043 < |w[168]| <0.045 0.051 < |w[169]| < 0.053 0.058 < |w[170]| < 0.060 0.065 < |w[171]|< 0.067 0.071 < |w[172]| < 0.073 0.077 < |w[173]| < 0.079 0.082 <|w[174]| < 0.084 0.088 < |w[175]| < 0.090 0.093 < |w[176]| < 0.095 0.097< |w[177]| < 0.099 0.101 < |w[178]| < 0.103 0.105 < |w[179]| < 0.1070.109 < |w[180]| < 0.111 0.112 < |w[181]| < 0.114 0.115 < |w[182]| <0.117 0.117 < |w[183]| < 0.119 0.119 < |w[184]| < 0.121 0.121 < |w[185]|< 0.123 0.123 < |w[186]| < 0.125 0.124 < |w[187]| < 0.126 0.125 <|w[188]| < 0.127 0.126 < |w[189]| < 0.128 0.126 < |w[190]| < 0.128 0.126< |w[191]| < 0.128 0.126 < |w[192]| < 0.128 0.127 < |w[193]| < 0.1290.126 < |w[194]| < 0.128 0.126 < |w[195]| < 0.128 0.125 < |w[196]| <0.127 0.124 < |w[197]| < 0.126 0.123 < |w[198]| < 0.125 0.122 < |w[199]|< 0.124 0.121 < |w[200]| < 0.123 0.119 < |w[201]| < 0.121 0.118 <|w[202]| < 0.120 0.116 < |w[203]| < 0.118 0.114 < |w[204]| < 0.116 0.111< |w[205]| < 0.113 0.109 < |w[206]| < 0.111 0.107 < |w[207]| < 0.1090.104 < |w[208]| < 0.106 0.102 < |w[209]| < 0.104 0.099 < |w[210]| <0.101 0.096 < |w[211]| < 0.098 0.093 < |w[212]| < 0.095 0.090 < |w[213]|< 0.092 0.087 < |w[214]| < 0.089 0.084 < |w[215]| < 0.086 0.082 <|w[216]| < 0.084 0.079 < |w[217]| < 0.081 0.076 < |w[218]| < 0.078 0.073< |w[219]| < 0.075 0.070 < |w[220]| < 0.072 0.067 < |w[221]| < 0.0690.064 < |w[222]| < 0.066 0.061 < |w[223]| < 0.063 0.058 < |w[224]| <0.060 0.055 < |w[225]| < 0.057 0.053 < |w[226]| < 0.055 0.050 < |w[227]|< 0.052 0.047 < |w[228]| < 0.049 0.045 < |w[229]| < 0.047 0.043 <|w[230]| < 0.045 0.040 < |w[231]| < 0.042 0.038 < |w[232]| < 0.040 0.036< |w[233]| < 0.038 0.034 < |w[234]| < 0.036 0.032 < |w[235]| < 0.0340.030 < |w[236]| < 0.032 0.028 < |w[237]| < 0.030 0.027 < |w[238]| <0.029 0.025 < |w[239]| < 0.027 0.023 < |w[240]| < 0.025 0.022 < |w[241]|< 0.024 0.021 < |w[242]| < 0.023 0.019 < |w[243]| < 0.021 0.018 <|w[244]| < 0.020 0.017 < |w[245]| < 0.019 0.016 < |w[246]| < 0.018 0.015< |w[247]| < 0.017 0.014 < |w[248]| < 0.016 0.013 < |w[249]| < 0.0150.012 < |w[250]| < 0.014 0.011 < |w[251]| < 0.013 0.010 < |w[252]| <0.012 0.009 < |w[253]| < 0.011 0.009 < |w[254]| < 0.011 0.008 < |w[255]|< 0.010 0.007 < |w[256]| < 0.009 0.007 < |w[257]| < 0.009 0.006 <|w[258]| < 0.008 0.006 < |w[259]| < 0.008 0.006 < |w[260]| < 0.008 0.005< |w[261]| < 0.007 0.005 < |w[262]| < 0.007 0.004 < |w[263]| < 0.0060.004 < |w[264]| < 0.006 0.004 < |w[265]| < 0.006 0.003 < |w[266]| <0.005 0.003 < |w[267]| < 0.005 0.003 < |w[268]| < 0.005 0.003 < |w[269]|< 0.005 0.002 < |w[270]| < 0.004 0.002 < |w[271]| < 0.004 0.002 <|w[272]| < 0.004 0.001 < |w[273]| < 0.003 0.001 < |w[274]| < 0.003 0.001< |w[275]| < 0.003 0.001 < |w[276]| < 0.003 0.000 < |w[277]| < 0.0020.000 < |w[278]| < 0.002 0.000 < |w[279]| < 0.002 0.000 < |w[280]| <0.002 0.000 < |w[281]| < 0.002 0.000 < |w[282]| < 0.002 −0.001 <|w[283]| < 0.001 −0.001 < |w[284]| < 0.001 −0.001 < |w[285]| < 0.001−0.001 < |w[286]| < 0.001 0.000 < |w[287]| < 0.002 0.000 < |w[288]| <0.002 0.000 < |w[289]| < 0.002 0.000 < |w[290]| < 0.002 0.000 < |w[291]|< 0.002 0.001 < |w[292]| < 0.003 0.001 < |w[293]| < 0.003 0.001 <|w[294]| < 0.003 0.001 < |w[295]| < 0.003 0.000 < |w[296]| < 0.002 0.000< |w[297]| < 0.002 0.000 < |w[298]| < 0.002 −0.001 < |w[299]| < 0.001−0.001 < |w[300]| < 0.001 0.000 < |w[301]| < 0.002 0.000 < |w[302]| <0.002 0.001 < |w[303]| < 0.003 0.001 < |w[304]| < 0.003 0.002 < |w[305]|< 0.004 0.002 < |w[306]| < 0.004 0.002 < |w[307]| < 0.004 0.003 <|w[308]| < 0.005 0.003 < |w[309]| < 0.005 0.003 < |w[310]| < 0.005 0.004< |w[311]| < 0.006 0.004 < |w[312]| < 0.006 0.004 < |w[313]| < 0.0060.004 < |w[314]| < 0.006 0.004 < |w[315]| < 0.006 0.004 < |w[316]| <0.006 0.003 < |w[317]| < 0.005 0.003 < |w[318]| < 0.005 0.003 < |w[319]|< 0.005 0.003 < |w[320]| < 0.005 0.002 < |w[321]| < 0.004 0.002 <|w[322]| < 0.004 0.002 < |w[323]| < 0.004 0.001 < |w[324]| < 0.003 0.001< |w[325]| < 0.003 0.001 < |w[326]| < 0.003 0.000 < |w[327]| < 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0.001 −0.001 < |w[603]| < 0.001 −0.001 < |w[604]| <0.001 −0.001 < |w[605]| < 0.001 −0.001 < |w[606]| < 0.001 −0.001 <|w[607]| < 0.001 −0.001 < |w[608]| < 0.001 −0.001 < |w[609]| < 0.001−0.001 < |w[610]| < 0.001 −0.001 < |w[611]| < 0.001 −0.001 < |w[612]| <0.001 −0.001 < |w[613]| < 0.001 −0.001 < |w[614]| < 0.001 −0.001 <|w[615]| < 0.001 −0.001 < |w[616]| < 0.001 −0.001 < |w[617]| < 0.001−0.001 < |w[618]| < 0.001 −0.001 < |w[619]| < 0.001 −0.001 < |w[620]| <0.001 −0.001 < |w[621]| < 0.001 −0.001 < |w[622]| < 0.001 −0.001 <|w[623]| < 0.001 −0.001 < |w[624]| < 0.001 −0.001 < |w[625]| < 0.001−0.001 < |w[626]| < 0.001 −0.001 < |w[627]| < 0.001 −0.001 < |w[628]| <0.001 −0.001 < |w[629]| < 0.001 −0.001 < |w[630]| < 0.001 −0.001 <|w[631]| < 0.001 −0.001 < |w[632]| < 0.001 −0.001 < |w[633]| < 0.001−0.001 < |w[634]| < 0.001 −0.001 < |w[635]| < 0.001 −0.001 < |w[636]| <0.001 −0.001 < |w[637]| < 0.001 −0.001 < |w[638]| < 0.001 −0.001 <|w[639]| < 0.001

The invention claimed is:
 1. An apparatus for generating audio subbandvalues in audio subband channels, comprising: an analysis windower forwindowing a frame of time−domain audio input samples being in a timesequence extending from an early sample to a later sample using ananalysis window function comprising a sequence of window coefficients toacquire windowed samples, the analysis window function comprising afirst number of window coefficients derived from a larger windowfunction comprising a sequence of a larger second number of windowcoefficients, wherein the window coefficients of the window function arederived by an interpolation of window coefficients of the larger windowfunction; and wherein the second number is an even number; and acalculator for calculating the audio subband values using the windowedsamples.
 2. The apparatus according to claim 1, wherein the apparatus isadapted to interpolating the window coefficients of the larger windowfunction to acquire the window coefficients of the window function. 3.The apparatus according to claim 1, wherein the apparatus or theanalysis windower is adapted such that the window coefficients of thewindow function are linearly interpolated.
 4. The apparatus according toclaim 1, wherein the apparatus or the analysis windower is adapted suchthat the window coefficients of the analysis window function areinterpolated based on two consecutive window coefficients of the largerwindow function according to the sequence of the window coefficients ofthe larger window function to acquire one window coefficient of thewindow function.
 5. The apparatus according to claim 1, wherein theapparatus or the analysis windower is adapted to acquire the windowcoefficients c(n) of the analysis window function based on the equation${{c(n)} = {\frac{1}{2}\left( {{c_{2}\left( {2\; n} \right)} + {c_{2}\left( {{2\; n} + 1} \right)}} \right)}},$wherein n is an integer indicating an index of the window coefficientsc(n), and c₂(n) is a window coefficient of the larger window function.6. The apparatus according to claim 5, wherein the apparatus or theanalysis windower is adapted such that the window coefficients c₂(n) ofthe larger window function obey the relations given in the table inAnnex
 4. 7. The apparatus according to claim 1, wherein the analysiswindower is adapted such that windowing comprises multiplying thetime−domain audio input samples x(n) of the frame to acquire thewindowed samples z(n) of the windowed frame based on the equationz(n)=x(n)·c(n) wherein n is an integer indicating an index of thesequence of window coefficients in the range of 0 to T·N−1, wherein c(n)is the window coefficient of the analysis window function correspondingto the index n, wherein x(N·T−1) is the latest time−domain audio inputsample of a frame of time−domain audio input samples, wherein theanalysis windower is adapted such that the frame of time−domain audioinput samples comprises a sequence of T blocks of time−domain audioinput samples extending from the earliest to the latest time−domainaudio input samples of the frame, each block comprising N time−domainaudio input samples, and wherein T and N are positive integers and T islarger than
 4. 8. The apparatus according to claim 1, wherein theanalysis windower is adapted such that the analysis window functioncomprises a first group of window coefficients comprising a firstportion of the sequence of window coefficients and a second group ofwindow coefficients comprising a second portion of the sequence ofwindow coefficients, wherein the first portion comprises less windowcoefficients than the second portion, wherein an energy value of thewindow coefficients in the first portion is higher than an energy valueof the window coefficients of the second portion, and wherein the firstgroup of window coefficients is used for windowing later time−domainsamples and the second group of window coefficients is used forwindowing earlier time−domain samples.
 9. The apparatus according toclaim 1, wherein the apparatus is adapted to using an analysis windowfunction being a time−reversed or index-reversed version of a synthesiswindow function to be used for the audio subband values.
 10. Theapparatus according to claim 1, wherein the analysis windower is adaptedsuch that the larger window function is asymmetric with respect to thesequence of window coefficients.
 11. An apparatus for generatingtime−domain audio samples, comprising: a calculator for calculating asequence of intermediate time−domain samples from audio subband valuesin audio subband channels, the sequence comprising earlier intermediatetime−domain samples and later time−domain samples; a synthesis windowerfor windowing the sequence of intermediate time−domain samples using asynthesis window function comprising a sequence of window coefficientsto acquire windowed intermediate time−domain samples, the synthesiswindow function comprising a first number of window coefficients derivedfrom a larger window function comprising a sequence of a larger secondnumber of window coefficients, wherein the window coefficients of thewindow function are derived by an interpolation of window coefficientsof the larger window function; and wherein the second number is even;and an overlap-adder output stage for processing the windowedintermediate time−domain samples to acquire the time−domain samples. 12.The apparatus according to claim 11, wherein the apparatus is adapted tointerpolating the window coefficients of the larger window function toacquire the window coefficients of the window function.
 13. Theapparatus according to claim 11, wherein the apparatus is adapted suchthat the window coefficients of the synthesis window function arelinearly interpolated.
 14. The apparatus according to claim 11, whereinthe apparatus is adapted such that the window coefficients of thesynthesis window function are interpolated based on two consecutivewindow coefficients of the larger window function according to thesequence of window coefficients of the larger window function to acquireone window coefficient of the window function.
 15. The apparatusaccording to claim 11, wherein the apparatus is adapted to acquire thewindow coefficients c(n) of the synthesis window function based onequation${{c(n)} = {\frac{1}{2}\left( {{c_{2}\left( {2\; n} \right)} + {c_{2}\left( {{2\; n} + 1} \right)}} \right)}},$wherein c₂(n) are window coefficients of a larger window functioncorresponding to the index n.
 16. The apparatus according to claim 15,wherein the apparatus is adapted such that the window coefficient c₂(n)fulfill the relations given in the table in Annex
 4. 17. The apparatusaccording to claim 11, wherein the synthesis window is adapted such thatthe windowing comprises multiplying the intermediate time−domain samplesg(n) of the sequence of intermediate time−domain samples to acquire thewindowed samples z(n) of the windowed frame based on the equationz(n)=g(n)·c(T−N−1−n) for n=0, . . . , T·N−1.
 18. The apparatus accordingto claim 11, wherein the synthesis windower is adapted such that thesynthesis window function comprises a first group of window coefficientscomprising a first portion of the sequence of window coefficients and asecond group of window coefficients comprising a second portion of thesequence of window coefficients, the first portion comprising lesswindow coefficients than the second portion, wherein an energy value ofthe window coefficients in the first portion is higher than an energyvalue of the window coefficients of the second portion, and wherein thefirst group of window coefficients is used for windowing laterintermediate time−domain samples and the second group of windowcoefficients is used for windowing earlier intermediate time−domainsamples.
 19. The apparatus according to claim 11, wherein the apparatusis adapted to using the synthesis window function being a time−reverseor index-reversed version of an analysis window function used forgenerating the audio subband values.
 20. The apparatus according toclaim 11, wherein the synthesis windower is adapted such that the largerwindow function is asymmetric with respect to a sequence windowcoefficients.
 21. A method for generating audio subband values in audiosubband channels, comprising: windowing, by an analysis windower, aframe of time−domain audio input samples being in a time sequenceextending from an early sample to a later sample using an analysiswindow function to acquire windowed samples, the analysis windowfunction comprising a first number of window coefficients derived from alarger window function comprising a sequence of a larger second numberof window coefficients, wherein the window coefficients of the windowfunction are derived by an interpolation by window coefficients of thelarger window function; and wherein the second number is an even number;and calculating, by a calculator, the audio subband values using thewindowed samples, wherein at least one of the analysis windower and thecalculator comprises a hardware implementation.
 22. A method forgenerating time−domain audio samples, comprising: calculating, by acalculator, a sequence of intermediate time−domain samples from audiosubband values in audio subband channels, the sequence comprisingearlier intermediate time−domain samples and later intermediatetime−domain samples; windowing, by a synthesis windower, the sequence ofintermediate time−domain samples using a synthesis window function toacquire windowed time−domain samples, the synthesis window functioncomprising a first number of window coefficients derived from a largerwindow function comprising a sequence of a larger second number ofwindow coefficients, wherein the window coefficients of the windowfunction are derived by an interpolation of window coefficients of thelarger window function; and wherein the second number is even; andoverlap-adding, by an overlap-adder output stage, the windowedtime−domain samples to acquire the time−domain samples; wherein at leastone of the synthesis windower, the calculator and the overlap-adderoutput stage comprises a hardware implementation.
 23. A non-transitorystorage medium having stored thereon a program with a program code forexecuting a method for generating audio subband values in audio subbandchannels, the method comprising: windowing a frame of time−domain audioinput samples being in a time sequence extending from an early sample toa later sample using an analysis window function to acquire windowedsamples, the analysis window function comprising a first number ofwindow coefficients derived from a larger window function comprising asequence of a larger second number of window coefficients, wherein thewindow coefficients of the window function are derived by aninterpolation by window coefficients of the larger window function; andwherein the second number is an even number; and calculating the audiosubband values using the windowed samples, when running on a processor.24. A non-transitory storage medium having stored thereon a program witha program code for executing a method for generating time−domain audiosamples, the method comprising: calculating a sequence of intermediatetime−domain samples from audio subband values in audio subband channels,the sequence comprising earlier intermediate time−domain samples andlater intermediate time−domain samples; windowing the sequence ofintermediate time−domain samples using a synthesis window function toacquire windowed time−domain samples, the synthesis window functioncomprising a first number of window coefficients derived from a largerwindow function comprising a sequence of a larger second number ofwindow coefficients, wherein the window coefficients of the windowfunction are derived by an interpolation of window coefficients of thelarger window function; and wherein the second number is even; andoverlap-adding the windowed time−domain samples to acquire thetime−domain samples, when running on a processor.